What is the proper name for these cheating F#m & G#m chords?
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It's basically a standard F#m barre, but without the barre. The low E string is generally not played. You can also move it two frets up to get a G#m shape, which still works but is not quite as nice - I can use this F#m in virtually any song but the G#m sometimes works, and sometimes doesn't depending on the song.
How would these be properly named?
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A form of F♯m11. As there's F♯, C♯, A, the triad of F♯m, with a ♭7 (E) and a 4 (B). 11ths sometimes have the 9th missing, as in this case, but must have a 7th of some sort. Can't be called a sus 4, as the 3 is still there. Often guitar needs to omit something, as it's not easily available to finger/fret. The ‘G♯m’ is an E maj7, and sounds better voiced with the bottom E, and without the top E.
To find the names of chords, establish a basic triad—usually maj. or min., then count up from the root to any extra notes. These give the number to attach to the chord name.
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