It sounds like Barenboim plays the ornament as a four-note trill starting on the auxiliary (i.e., D♭-C-D♭-C), and Kissin plays it as a five-note trill starting on the main note (C-D♭-C-D♭-C).
There are a couple of reasons you may hear differing interpretations here (and elsewhere), including a third possibility already mentioned: a three-note inverted mordent (C-D♭-C).
First, there's no hard-and-fast rule about whether Chopin's trills should begin on the main note or the auxiliary. Sometimes one has a sense of which is intended from the context, sometimes a manner of execution has become accepted as standard performance practice, sometimes one guesses.
Second, Chopin frequently used the tr. sign and the zigzag sign of an inverted mordent interchangeably. If tempo and duration of the note permit, tr. might indicate an extended trill with more than one alternation; on a note of relatively small value at fast speed, it's more likely to be the equivalent of an inverted mordent simply because, as a practical matter, there isn't time for a more elaborate ornament.
In short, there's latitude to play the trill in a number of slightly different ways. The most important thing to know is that it must begin on the beat, regardless of the number of alternations or whether you start on the main note or auxiliary. Playing the ornament such that it ends on the beat (e.g., C-D♭-C where the last C coincides with the bass note) would not be stylistically correct (although you may encounter that, too, in a professional performance by a pianist who's not well-informed).
Topic: Chopin Op 64 no 1 Trills