I saw/looked at the girl singing on the stage

a. I saw the girl singing on the stage.

b. I looked at the girl singing on the stage.

Does (a) mean "I saw the girl who was singing on the stage" or "I saw a scene where the girl was singing on the stage" or either (depending on context)?

Similarly, does (b) mean "I looked at the girl who was singing on the stage" or "I looked at a scene where the girl was singing on the stage" or either (depending on context)?

In case the questions are unclear, the above questions can be rephrased in terms of syntax as follows:

In (a), is singing on the stage complement of the noun girl, thereby constituting the NP the girl singing on the stage, or is it complement of the verb saw, or is it either (depending on context)?

Similarly, in (b), is singing on the stage complement of the noun girl, thereby constituting the NP the girl singing on the stage, or is it complement of the verb looked at, or is it either (depending on context)?

I saw the girl singing on the stage could mean that you witnessed her performance, as in I saw Kenneth Branagh play Hamlet.

It could also mean that she just happened to catch your eye. I saw the girl singing on the stage but was too preoccupied to notice what she was wearing.

In I looked at the girl [who was] singing on the stage, the complement identifies which girl was being looked at.

Your rewrites of each mean pretty much the same thing as the originals. Whether you say it or not, we can assume that if a girl is in a scene, it was a scene you saw or looked at her in.

The only real difference is the definitions of see and look at.

When you say you see something, we usually undertand that you're observing or noticing it for the first time:

I see that you're happy. I saw you drive by my office yesterday. Next time I see a red light, I promise I'll stop.

When you look at something, it's a single action of directing your eyes at something or someone:

Every time I look at you, you seem more worried. I looked at the girl on the stage and noticed her shoes. Look at me when I'm speaking to you. I can't look at him without thinking of last year's Christmas party.