The word "look" is typically a verb: in its infinitive form, it is "to look".
I want you TO LOOK at me.
But in your sentences, "a look" is acting as a noun phrase. It encapsulates more than just the act of looking. Instead, it actually deemphasizes the act of only seeing, and instead suggests an opportunity to interact.
"I want a look that problem when you're done."
Say, "I want to look out loud" 5 times fast.
You'll hear that it could be heard as, "I want A look", because the "to" and the "a"
(I hear it often as "I wan-tahl-ook".) Think French Liaisons. The "want to" requires a distinct double "t", and "want a" only has a single "t". If one is not precise about the enunciation, the difference in phonetics can become blurred.
So, the solution to this is to add a separator between the "want" and the "to". When the emphasis is on "to look", the phrase "to have" will be added so that you can clearly hear the "t" in "to".
When it's focusing on the "opportunity", "want" can be immediately followed by "to" (that is, "I want to look"). But this pattern will sound as if the individual is saying " I want a look" regardless of whether they mean a look or "to look")
This speech pattern will echo itself in writing, and thus we have the distinction between 1 and 2, with 2 being the clearer indication of the "a look" phrasing.
3 and 4 don't have that same issue, so there should be less of a discrepancy between them.
It is clear that 3 is Sub. V. Do. " I want a look" with "closer modifying look, but removing the ambiguity of the phrasing. Example 4 just ends up being less efficient.