I'm pretty sure that a sinousoidal sound wave actually registers as exactly one pitch in the ear. Other periodic functions can be expressed as an infinite sum of sinousoidal functions each of which has a frequency that's a multiple of the frequnency of the periodic function plus a constant function. I heard that the range of frequencies humans can hear goes from 20 to 20,000 Hz. That's probably just approximate for most people. I'm guessing that that range is true only for the sinousoidal tone. If a 10 Hz sinousoidal sound wave is played you probably can't hear it but if 10 Hz of another tone is played, you might hear it and it might sound very wierd because if you express it as an infinite sum of sinousoidal waves, you can't hear the sinousoidal wave that's 10 Hz. I've noticed that I could not sense the relative pitch of very low notes on the piano compared to higher notes. That might be the reason. Maybe the reason is because hair cells in the ear for very low notes register a wider range of frequencies. I think the same might also be true for me about really high notes that I can't sense relative pitch of them. Also, I believe the math shows that for sound whose frequency is more than half the highest frequency sinousoidal wave that you can hear, you can't distinguish between one periodic sound wave of that frequency and another periodic sound wave of the same frequency if the latter sound wave is changed in amplitude to sound the same volume.
I think that for almost all people who have absolute pitch, of the pitches they can sense relative pitch for, if they can sense absolute pitch for one of those pitches, they will be able to figure out how to sense absolute pitch for any of those pitches. I don't know that for sure. If that's so, then maybe people didn't bother inventing a very precise definition of absolute pitch for that reason.