I have trained in classical piano for 14 years, but I'd love to improvise to play jazz piano...very similar to the above video: calm, melancholic, quiet, smooth jazz.
Where can I get started? Do I need to memorize a whole book of jazz chords?
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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.
learn how harmony works - chords and chord progressions - and how to reharmonize chord progressions, because most jazz progressions are actually reharmonizations of what might otherwise be duller, simpler progressions.
learn chord voicings for those chords
Start learning how to play by ear instead of using sheet music
find a pianist who plays jazz piano (professionally) in your community and ask for private instruction
join a band
The The Jazz Piano Book by Mark Levine is kind of a reference for jazz standards.
I learned starting by open and close positions for major, minor and dominant chords as shown in the book.
Then practice playing II-V-I combinations in 5ths (that is pretty much a Jazz standard) alternating between closed-open-closed exercises and open-closed-open exercises. Getting to this stage will make you feel you have the jazz sound to play around.
I quite recommend the http://www.jazzadvice.com/ site for very interesting explanations of jazz piano in general.
Also my teacher website: http://www.jazzpiano.co.nz