To general public "keyboard" (as musical instrument) would be "синтезатор". To those who know what they are talking about they'd rather call it "клавиатура" (well, likely so). When I decided to learn music a bit and buy a "keyboard" to me it was naturally "синтезатор", but then I gradually realized that it was closer to "клавиатура", and "synthesizer" has more to do with synthesizing sounds, something like a Moog synthesizer. Btw, I think "клавиши" is more concerned with the act of playing, so if you need to say something like "he is on the keyboard" it's better to use "он на клавишах", as Denis noted.

Usually it's just ''клавиши''. Eg.: играть в группе на клавишах; купить клавиши.

My bro is a musician. He says that a piano's keyboard is клавиатура. Синтезатор is a musical instrument.

Yes,you are right.

It's a great tip.I've never noticed that left side before.I'm a right-handed person.:)