American Heritage Dictionary explains protagonist as coming from the Greek word protagnists, which is the Greek combining form proto-, meaning first in rank or first in time, plus the root agnists, meaning actor. So, the definition of protagonist to work with, so as to recognize a protagonist, is that a protagonist is the character (or sometimes characters) who is the first in importance and, sometimes, the first presented. How does this fit in with Hemingway's "A Clean, Well-Lighted Place"?
Hemingway starts by drawing attention to "an old man who sat in the shadow the leaves." While he is the first in time, being the first individual mentioned, he is not first in importance. He does, nonetheless, act as a catalyst for the inner conflict, the self versus self conflict of the protagonist, who is the waiter and the next character mentioned. The waiter is the at once a second narrative voice (""Last week he tried to commit suicide," one waiter said. ... "He was in despair.") and the voice through whom the conflict is analyzed and described, thus giving him greatest importance.
It was a nothing that he knew too well. It was all a nothing and a man was a nothing too. It was only that, and light was all it needed and a certain cleanness and order. Some lived in it and never felt it, but he knew it all was nada y pues nada y nada y pues nada.
The young waiter can't qualify as the protagonist--even though it is his life we are given most intimate details for--because he leaves while the first waiter continues his conversation by himself. He speaks to himself of existential angst, about despair, about nada, and about the importance of a clean, well-lighted place that lends to the illusion of keeping the inner conflict at bay. For these reasons, the waiter who hopes his sleeplessness is only insomnia, is the protagonist.
A clean, well-lighted cafe was a very different thing. Now, without thinking further, he would go home to his room. He would lie in the bed and finally, with daylight, he would go to sleep. After all, he said to himself, it's probably only insomnia. Many must have it.
I belive it is the older waiter, who do not have a name is the protagonist.For the old man, and indeed, the older waiter, the café represents a refuge from their despair. It is a place where they can usually linger for as long as they please. For this reason, the older waiter prefers to keep the café open as long as possible in the event there is someone who will "need a light for the night."