training program with several puck.js

is it possible to do all this with puck.js lite or do I have to use the v2 version?

Yes, the Lite is fine as all you're using is the button.

You'd need another Espruino device for the master device to have it run as-is, but you could for instance use a Pixl.js if you want a screen (https://www.espruino.com/Pixl.js), Bangle.js on your wrist (https://www.espruino.com/Bangle.js2) or just another Puck, and then arrange to connect your phone to that.

You could use one of the Pucks from the game as the master too - it's just the code would get a tiny bit more complex.