Is the ontological argument a logical proof?

Given that a being than which none greater can be imagined does exist in the mind, as an idea,
and given that a being that exists in reality is greater than a being that only exists in the mind,
does it follow that a being than which none greater can be imagined actually exists in reality?

Or does it only follow that the idea of a being than which none greater can be imagined
is the idea of a God that exists in reality?
First things first:

Does a being than which none greater can be imagined exist as an idea in your mind?

Presumably you would need to know what "greater" means in this context. Does it mean more powerful, wealthier, more popular, bigger, more tasteful, more politically correct, morally better, better looking, all of those and more, or just some of them? Perhaps the main thing here is that if there is a God in reality then we would be unable to imagine any greater being. Still, it is conceivable that God could imagine a being greater than Himself, because if there is a God then He may well be capable of feats much greater than any of us could imagine.
Second things second:

Is a being that exists in reality greater than a similar being that only exists in the mind?

Suppose that there had been a real Robin Hood, instead of our fictional Robin Hood. We might never have heard of that real Robin. And Robin Hood's greatness is presumably his power to inspire. So it is quite plausible that that hypothetically real Robin would not be greater than the fictional Robin that we know and love. Still, the main thing here is that a real God would presumably be greater than any fictional God.
So thirdly:

Could an ontological argument be a logical proof?

Surprisingly the answer is "yes": a very well-defined ontological argument could be a formal-logical proof that something called "God" has something called "existence" (and nowadays "logic" does usually means formal logic).