There isn't, couldn't , and shouldn't be a prescription for what to play as a piano player playing in a jazz ensemble.
Piano is unique - along with guitar - in that it can play single notes, chords, and be a percussive instrument, sometimes all simultaneously. (And even be the bass instrument!) As such, it can play in its role as part of the rhythm section rather well.
When someone is soloing, the piano playing can take on many different mantles. In the OP's example, it uses small chords, often syncopated. Almost a pared back route of the harmony of the piece, as if to remind the listener (and player) where they all are in the piece. Just little stabs, not getting in the way of the soloist, purely supporting. Dyads work well here, using, for example, 3rds and 7ths of the prevailing chords, as no more is needed. Giving the space that's required for some solos/soloists.
In other situations, bigger, wider chords work, as a wash under what's being played by the solo player. Or the pianist can work with the drummer or bassist - or both, to play about with the rhythm of the piece, using syncopation, stabs, etc.
Other times, the pianist might be having a conversation with the soloist, and/or throwing ideas - rhythmic and/or harmonic, out there for the soloist to pick up on, and react to in their playing.
There are so many different ways to be the piano player in an ensemble, depending on many factors - the song itself, what the make up of the band is, who's soloing, how they're soloing, and mst times, even if the piano player uses stabs, as mentioned earlier, they won't come in the same places as the guy put the last time he played. That's one of the facets of playing jazz, 'going with the flow' means each performance can be different and new.