Your definitions of "click" and "click on" seem spot on in my opinion. I would almost always say (there's always exceptions in the English language!) "please click the right mouse button" and "make sure you click on the menu icon".

It's personal preference. I think that I don't use the 'on' - and find it strange when I hear ads on TV that do use 'on'. It's English. It's a living language. There aren't necessarily hard and fast rules.

I completely agree with the above answers. Thought I would share some furthers insights A command is the most forceful way to tell someone to do something. Commands are more powerful the shorter they are. When you are training a dog, and trying to assert your power you tell him to "sit" /"stay" 'click this link' is more slightly more powerful than 'click on this link' (even though not much) Often we see 'click this link' in the context of advertising. Marketing relies on creating a powerful message. If omitting the on helps even a little, then this is what they will do. In turn this is what we become accustomed to seeing, and therefore continue to use it. As another point: You cannot say click on your fingers... but rather click your fingers (the same way as you wouldn't say -'bend on your arm) Maybe click (without on) is used commonly with regards to computers as a way of differentiating.

Hmm, so I guess there's no strict rule to it? After asking a few people around me, it does seem like it's more of a personal preference thing. I wonder if it's regional? Gary, Greg and I are all from different countries...