I am trained in classical piano for 14 years but I'd love how to improvise to play jazz piano...very similar to that video, calm, melancholic, quiet smooth jazz.
where can I get started? do I need to memorize a whole book of jazz chords?
When I asked a buddy of mine the same question from a very similar background (12 years of classical piano), he gave me two steps: Listen and Emulate, then Create. The idea being, find the sound you want to have. Then find musicians that play that sort of music and do everything they do. Transcribe solos, chords, even entire pieces. Do your best to copy their sound. Over time, your mind will start to pick out patterns and you'll be able to improvise with those sounds.
I would only recommend memorizing chords to the point where you can go from their sound to their name, and then play them. For instance, you should know how to build a C13 chord. If you have to build it up from the C triad, then count up to the 13th, that's perfectly fine for this process. It's far more important to get a feel for how the chords should fit together, and that happens through this process of listening and emulating. Over time, you'll learn the chord names through necessity.
I would recommend two books: