The glass and its content

The saying 'is the glass half-full or half-empty?' has always riddled me.
Let me explain.
It is supposed to describe a person with an optimistic or a pessimistic view. But what is what?
Suppose a person sitting at a bar with a half-full glass in front of her. She can contemplate on life in general, and on the fact that her glass is filled with half of its original content.

That means that (a) she has emptied half of it, and (b) she still has the other half to enjoy. Both of which would, if I were that person, be good, and a reason for happiness.
Now consider a person sitting at a bar with a half-emptied glass in front of him. He would contemplate on this same theme, coming to the same conclusions, namely that (a) he has already enjoyed half of its contents and (b) that there is still another half to go.
No matter from which angle I approach the issue, I can't see a negative side of the situation. Which is why I find this saying so hard to interpret.

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