As I understand it, when this generator is used to play a musical piece, the correct term for the part of the piece produced by the white noise generator is "percussion".
Not really - 'percussion' refers to sounds that are produced through something being hit. Of course when you are synthesizing sounds, nothing is really being hit, but a percussive synthesised sound might mean something with a fast attack. You can have percussive sounds that are still pitched, such as a xylophone or glockenspiel.
As an everyday word, I would just say noise/noisy or noise-based sounds. That's generally understood to refer to a sound that contains energy spread over the frequency spectrum (or part of it), rather than being concentrated in particular partials/harmonics.
The problem with saying 'unpitched' for these noisy sounds is that you can have sounds with definite, non-integer multiple partials that still don't sound pitched, and yet don't sound noisy either (and wouldn't be generatable with a noise generator). These often sound clangy or metallic - not pitched, but not noisy either.
Similarly, the parts of the piece produced by the sine wave generators are "melody" and "harmony". Is there a single term for the combination of these two - a single term for the parts of the piece that can be reproduced as sine waves? Perhaps the "tuned" or "pitched" parts?
If you are using your sine waves to produce harmonic spectra, then you are indeed producing sounds that will be perceived as having an identifiable pitch, so pitched is a good term to use.
However, you can use sine waves to produce clangy/metallic unpitched sounds, and if your sine generator is efficient enough to produce a large number of sine waves, it will be possible to generate noise-like sounds (including white noise sounds) only with sine waves - no noise generator needed.
Another way of looking at this is that actually the separation of sounds into these categories of pitched / unpitched / noisy is a little bit arbitrary, and the more you look into the nature of real sounds, you can't really draw clear lines between them. We tend to draw those distinctions because it's easier to think about sound like that from a beginner perspective, and if an additive synthesizer is not very powerful, it makes sense from a practical point of view to help it out with a noise source. But there's really no clear dividing line between a sound made of sines and a noisy sound: 'noise' is the equivalent of lots of sine waves at different frequencies, and a sine wave isn't so different from very narrow-band noise.