The thing with the precision is hard to describe (and hell hard to implement - I can imagine). But overall I thing it might be a little delayed audio what causes it. I re-run second track on medium difficulty to check and notice that at the first part I would expect some beats to match the piano note. Later with the percusions beat introduced - I would expect it on the beat. In both cases I had to push key a little bit earlier.
But once again, it might be totally my problem that I am fixing on "wrong" flow.
EDIT: I also tried easy difficulty on the same track and it seemed on point there. So either it is little off for medium on that track or my brain is overwhelmed with more inputs and that is what creates this feeling.