This is a very complex subject. Let me see if I can add to the confusion.
Your piano has a number of sampled sounds hard-wired into its memory. The sample was created by playing a note and recording it. Better samples record it at several velocities, let it ring longer, record more notes without interpolating sounds (up to 88 for a piano) etc. The note is the full sound spectrum.
The piano's software tweaks these samples in a variety of ways to create the instruments you get when you press the appropriate button. Some of the tweaking can use up more than one note of polyphony. Layering is an example of this. The only way I know to tell if an instrument is layered is to read the manual or look at the programming. Other tweaks don't consume polyphony - they add reverb or other effects, change the sound's envelope, etc.
For a basic piano sound, the usual relationship is one note = two voices of polyphony because the sounds are sampled in stereo. So if you play a 4-note chord you'll use up 8 voices. If you hold the sustain pedal down and then play another 4-note chord you're now using 16 voices because the first 8 are still sounding. Get the idea?
So does a choir use lots of voices? Not necessarily. If they sampled, say, a full choir singing in stereo, then each note might only take 2 voices. But if they sampled soprano, alto, tenor and bass voices and then layered them for a "full" choir you might use 4x2=8 stereo voices per note.
Most of this has very little to do with the "advanced" technology of the piano or lack of it. Better instruments typically have bigger and better samples so the sound is richer, lasts longer, whatever, but the basic polyphony issue remains 1 note = two voices. Usually.
Note-stealing happens when the piano runs out of polyphony. Different manufacturers handle this differently. The easiest algorithm is "first in, first off" - that is, the oldest sound is cut off first. There are other ways of doing it that are more complicated and may take more processing power, so I guess that's a place where faster computers help. Modern note-stealing techniques are pretty good.
Bottom line: more polyphony is better; 64 notes is probably good enough for solo piano; complex orchestrations want more than that; note stealing works OK for most things.[/b]