: What is a definition of "oscillator" that is suitable for all musical instruments? I understand from my music theory class that for reed and brass instruments, the "primary oscillator" is the modulation of an airstream. Comparatively,

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The common factor in all wind instruments is that sound is produced from a vibrating column of air, set into oscillation by a player's breath. But the air column isn't the "oscillator", it's the thing that is made to oscillate. So I think your theory class has got it a bit wrong! There are three ways in which a player may be the "oscillator". In a brass instrument he "buzzes" his lips into the mouthpiece. In a reed instrument the reed(s) take over the function of the lips. In a flute it's rather different, oscillation is the result of an airstream splitting in two when it hits an edge. So three different ways of being an oscillator, all with the result of getting an air column vibrating, ready to be shaped by the rest of the instrument.
The flute family produce a waveform reasonably like the classic sine-wave picture, with an easily discernable frequency and amplitude. Brass and woodwind have much more complex waveforms. And a large part of the sound's characteristic, for all of them, is defined by the attack, where the waveform looks a LOT less tidy. As so often in musical topics I think you'll be better off learning how individual instruments do work, rather than searching for a Unified Theory. Maybe your teachers are simply looking for you to say "wind instruments work by the player setting an enclosed air column into vibration".