I only remember there being one instance of trills in the third movement (two instances, really, but it's the same material), so I hope I'm addressing the right ones.
Beethoven's ornaments are often hard to formalize because he came right at the time when the common practise was changing from starting on the higher note to starting on the principal note. Common wisdom is that the earlier sonatas are played in the baroque style of ornamentation, and the later ones in the romantic style. This also doesn't help, because the tempest is right in the middle. So, we have to go to the music itself.
In the second movement, the melody before the trill is not technically in the voice that the trill occurs in. There are three Cs in a row in that top voice; the music one pays attention to is in the changing harmonies beneath them. Thus, the player wants to maintain the integrity of that C as long as possible in the top voice to make the lower voices more evident. In other words, it simply makes sense to start on the C rather than the D.
In the third movement, the melody (without the ornamentation, as is evident in the next phrase where it's in octaves) is a simple fall from F to E, which makes it a descending line. If you started the trill on F and went up, it would ruin the direction of the line, making it go up a bit, then back down; very confusing for a listener to follow in such a short space. Starting on the G, then, the line simply goes straight down, and the initial G is interpreted as a pick up note to the F. This trill is actually to create dissonance (B-diminished chord with a G) which resolves to the harmony of E major, rather than simply ornament the melody. In this case, because the G is sort of part of the harmony, you want to keep the trill going as long as you can before resolving it downward, but remember to make keep it rhythmic and in-time, even if that means not trilling as fast as you possibly can. This should have a slightly-classical feel to it, rather than a full-blown romantic tremolo sound.
The other trill, coming a measure after the ones on F, is again different. This time it occurs in the middle of a melodic fragment; specifically, a melody that is moving in steps. If you start that particular trill on D, it would be repeating the previous melody note, and the integrity of the line would be ruined. Starting on the principal note here continues the melody downwards, and thus is should be seen as a baroque-style ornament, rather than part of the harmonic texture. It then also follows that, in this case, it is best to make just one turn to the D, rather than trilling for the entire duration of the note.