: How to visualize scales? I am trying to become a more versatile musician, and I am pretty sure being able to in some way imagine where on a scale the notes, that I am playing are located is an important part of that. This

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In this video, guitar scales are visualized, quite literally, in different ways, from simple to complex: as plain dots showing the scale shapes, colored dots showing chord notes, and finally as scale/mode degrees.

I'm linking to the point in the video where the scale degrees are visualized, since that's what you asked specifically, but if you want to get to the bottom of the whole issue you should probably take the time to watch the whole thing from the beginning. (It's not easy and it's not short, but I think it will answer pretty much everything you asked in the question)

For full disclosure, I worked on designing the software shown in the video, which is called the Guitar Scales Method.

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EDIT : this answer was made before any edits to the question!

Knowledge of at least some of the scale patterns is your answer. On guitar, there is the unique factor that doesn't occur on any other instruments, except bass guitar.

If you know just one scale pattern, let's say the two octaves that start on 6th string fret 5, then 7, then 5th string, fret 4,5,7 etc - giving an A major scale across all six strings. By moving that same pattern up a fret, you have the scale notes freom B♭ major, and so on.

Now, the degrees of the scale are viewed from a different perspective. Let's say you play notes 1, 3 5 in key A. That's 6th string 5th fret, 5th string 4 and 7th frets. To play the same tune in B♭, just move iyt all up one fret. To play in key C, start on 6th string, 8th fret, but retain the same relative pattern.

So once you establish a tune in one key, it's straightforward to transpose to other keys.

When you are familiar with that pattern, find a different one, and work it similarly. Soon, you'll be able to hear the different intervals, which relate to parts of a pattern. Then, you'll be able to transfer a phrase from one pattern to another.

It may be easier to start with the pentatonics, as they form easy patterns on guitar necks.

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I'm still not convinced your question makes much sense. For one thing, music is loaded with accidentals and modulations (effectively key changes or major/minor stuff). Rather than worrying about where you are in the tonic scale, try to think about the current phrase and the absolute intervals between notes. It certainly helps to know that, e.g., the next chunk of notes are a run in D-melodic minor, but overall what you need to be able to do is "know" the melody and chord structure in your head. And without consciously assigning Solfege or chord names/numbers as you go.

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I'm going to tell you how I approach playing, soloing, scale studies, and just about everything musical. First off I study a lot, an awful lot. Everything I study, I try to get to the point that it becomes automatic, kind of like speaking. If a thought comes into my head that I wish to communicate, the words are automatically there for me and I can say them automatically. That's also the way I play my music. I can hear another person in the band play something that inspires a response in me and so I play the response automatically. It can sound like everyone's talking at the same time, or it can come out sounding like a more polite conversation, depending how courteous the band members are. One point I'd like to make is that along with the automatic response that I try to achieve, I also try to keep it relatively simple and straight forward. When I play, I want to convey feelings and emotions, paint invisible pictures with the sound of my instrument. I can do that because I know what I can trust my hands and my instrument to do and they do it automatically. It wasn't always the case for me, but I kept at it and I found it happening for me. That's just my personal experience. I'm adding an afterthought here: Somewhere in my studies I came across several pages of Melodic Patterns. These are exercises that I practiced until they became automatic for me. I can hear these patterns in my head and play them as if I were speaking. I suppose you could probably find more information about them by googling "Melodic Patterns" for what ever instrument you're interested in

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I think what you're grappling with is the relative interval relationships and function of scale tones.

There are several ways to describe those interval relationships:

intervals above/below the tonic (the first tone of the scale) is probably the commonest
the interval size between steps of the scale (whole and half steps, and augmented seconds) this and the first are surely the most common ways to describe the intervals within a scale

Along with those two things you want to learn the scale degree nomenclatures like tonic, supertonic, mediant, etc. and solfege like DO, RE, MI, etc.

Some other ways to think about it:

scales are two tetrachords (4 tone, scale like segments) separated by a perfect fifth. There are four tetrachord types: major, minor, harmonic, phrygian. With a tonic of C the tetrachords would start on C and G. All the combinations of tetrachords starting on those two tones (major+major, minor+major, etc. etc.) will produce a wide variety of scale. The nice thing is a fairly simple idea - four tetrachords - produces lots of scales. You need a good amount of basic theory under your belt to really understand this. Think of it like scale study, phase II.
scales can be viewed as filling in a tonic chord. Start with a major or minor chord then label the chord tones with their scale degree numbers. Ex. C major, start with the tonic chord C E G C labeled with scale degree numbers ^1 ^3 ^5 ^1...
^1 steps up/down from ^2
^3 steps up/down from ^4
^5 steps up/down from ^6
the high ^ steps down/up from ^7

^1 ^3 ^5 ^1 plus the filling in tones ^2 ^4 ^6 ^7 gives you a complete scale. A nice bonus with this view is the tones are basically group into the stable tones of the tonic chord and tendency tones and their characteristic movements. This knowledge is valuable in harmony and voice leading. This also involves a basic theory foundation and might be a scale study, phase II topic.

I want to be able to not think a' cis'' e'' gis'' if I am playing the first bar of Saties Gymnopédie No. 1, but the fourth, the sixth, the first and the third degree of the scale (in this case e major) or at least represent them this way in my head and their relation to each other and the scale as a whole.

This is exactly what you should do. You want to think in relative term.

Applying some of the ideas above to Gymnopédie No. 1 (my copy is in D with a key signature of two sharps) you will see for example...

The first bass note G is not the tonic but the subdominant - the fourth scale degree - of D major. This is really important to understanding the harmonic gesture. It doesn't start on the tonic, it's a gentle fall from up at the subdominant down to the tonic. It's like a little sigh of relief that sets the whole mood. The point is we can make these observations about up/down until we know where tones lie in the tonality, in the scale.
The top note of the first two chords F# is the mediant. Again, this tone's identify in the tonality is important. First, it's the tone that determines whether the mode is major or minor! Also, melodies starting on or moving about the mediant often have a sweet, lyrical quality and that is apparent in this music.
Fast forward to the end and look at the mediant. It becomes F natural, a minor third above the tonic. That lends a certain gravity to the end. It gets kind of heavy, brooding. That plays with the special role of the mediant.
When the melody proper starts in bar 5, notice it starts on the mediant, but also notice that the melody outlines the tonic chord D major. That demonstrates the filling in the tonic chord idea. Also some of the tendency tone stuff can be seen: G moves down to F# (^4 to ^3) and C# moves up to D (^7 to ^1), there is also a B down to A (^6 to ^5) but the motions is "delayed" by two beats.

I'm not sure this is the thing you are struggling with, but it is what comes to my mind when reading your question.

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This seems to be a very guitar-centric question and you are asking for "visualization" so I'll give you some tips on that. But I will also deviate from your question.

Given that the guitar is tuned to standard tuning there are a set of pattern that essentially stay in position and cross all 6 strings (I should have stated that this is not in reference to the 7 string guitar, etc). These patterns interlock up the neck from one position to the other. Starting with the 7 diatonic modes the 5 that I find myself using most often are Locrean, Dorian, Phrygian, Mixolydian, and Aeolian (minor). The reason for this affinity is that they also line up with a chord form from the CAGED system. This covered the entire terrain of the neck. As an example the Locrean mode can be played with the following finger pattern with one finger per consecutive fret,

string 6 --- 1, 2, 4

string 5 --- 1, 2, 4

string 4 --- 1, 3, 4

string 3 --- 1, 3, 4

string 2 --- 2, 4

string 1 --- 1, 2, 4

The second octave is at finger 1 on string 1. Starting on the second note you have the Ionian mode or major scale. The pattern does NOT depend on where you start. It is a movable pattern and once mastered the guitarist can read and play in any key just by aligning Do, the second not of Locrian with the appropriate fret (corresponding to the key name). For example, you want to play in Db just make sure that your index finger is one the 8th fret (C) and this will be C Locrian = Db Major. Similar patterns exist for the other modes. Once you have the standard forms down and understand how they overlap you can weave in and out of them all. It is a common practice on the guitar at least to commit 3 note per string patterns to memory as they lend themselves to fast consecutive picking patterns. These typically shift away from the original position. A third way to "visualize" scale patterns on the guitar is to realize that the strings can be grouped into pairs, (E, A), (D, G), (B, E), and that everything repeats up an octave when you shift 2 frets from group 1 to group 2 and then shift by 3 frets from group 2 to group 3. Yet another way to visualize is to see the whole step half step pattern one a single string. The pattern will need to be shifted for each new key.

Scales are built from simpler objects called tetrachords. The major scale, for example is the same tetrachord played once at a starting note then repeated a whole step up from where you stop, e.g. in terms of intervals (W - W - H) - W - (W - W - H). Additionally the 7 diatonic modes are all related to the Major scale. Hence, once you have all the basic 4 note patterns in your muscle memory and develop a good ear you can navigate you way around the neck without the visualization aspect associated with scales. Some would consider that a much better approach to fret board mastery.

Since you asked what we do as part of your question I will answer that directly. My guitar teacher encouraged me to "visualize" the scale, and chord, patterns on the fret board, and I did. At this point I "feel" my way around. I don't really think I use visualization when I play. I do sometimes close my eyes and visualize the sheet music if I'm still trying to memorize a piece but truthfully once that happens, or if I'm improvising, I go from sound to movement. I cannot say that the visualization exercises I was taught did or didn't help w/r to developing the ability to feel my way around the guitar. I believe it did help.