My piano teacher and I have been wondering how to play the trills, especially the second trill. Could you guys also tell me why the trill is played that way? Thanks.
There is a good deal of room for interpretation in how trills are played, for example how many repetitions of the trill notes you should use in a given example, whether you start a little before, right on, or a little after the beat, whether you start on the upper or lower note, and so on.
Different things factor into why a trill is played the way it is played: what the performer wants to do, what the performer believes is a correct interpretation of the trill, and what the performer can do. Better pianists can fit more notes in on a trill, although they may not want to.
Here are several different performances, each doing the trill differently. You can hear them well if you lower the speed to half (hit the little gearbox icon and you'll see the Speed menu).
Here's de Larrocha, using four notes, starting on the upper one (to answer your title question, this is the way I do it as well):
This one starts the trill on the lower note, with five notes total:
Here's Lang Lang doing a characteristically eccentric grace note on the note before the trill, and then starting the trill a few notes early:
This one does a note plus a turn; holds the first note of the trill (lower note) for twice as long as the others, then an upper and a lower:
Here's Sokolov just showing off. I can't quite hear how many notes he fits in, even at half speed (I think it's nine, started a little before the beat, and playing the following 16th notes at the same speed as the trill):
In this one, a very young Yuja Wang simply treats the trill as a mordent (lower-upper-lower):
This is the easiest way to do it. It's perfectly acceptable, so if you can't fit more notes in, I'd suggest that you go with it. However, if you really want five notes and can't fit them in in the time you have available, you can "cheat" by playing the first two notes of the trill ahead of the beat, like this (Uchida does this as well):
Sounds great, and it's hard to tell that you're "robbing Peter to pay Paul." :)
I wouldn't learn to play Mozart by listening to people who are making it up as they go along. We have learned a lot about articulation, phrasing, ornamentation and embellishments in Mozart through studying the writings of his contemporaries, and through careful examination of Mozart's own notation. I would start with Eva and Paul Badura-Skoda's book "Interpreting Mozart at the Keyboard". And beware not just of pianists who are making it up as they go along, but also of old editions which can play havoc with Mozart's notation.