Technically, Hemingway's "A Clean, Well-Lighted Place" is written, for the most part, with the point of view of the omniscient narrator: it is this narrator who is talking to the reader. However, since all but three or four paragraphs are dialogue, two other perspectives come from the contrast of the two waiters, one of whom is young, the other of whom is an older man, who is more sympathetic to the old customer.
And, near the end of the story, after the younger waiter departs, the point of view does, indeed, switch to that of the older waiter, who remains. This point of view then is first person narrator: for that time, it is the older waiter who is talking to the reader:
...Turning off the electric light, he [the older waiter] continued the conversation with himself. It is the light of course but it is necessary that the place be clean and pleasant. You do not want music. Certainly you do not want music. Nor can you stand before a bar with dignity although that is all that is provided for these hours. ["You"= a person: here the older waiter is just thinking, not talking to anyone in particular.]
However, this first person narrator switches back to the omniscient narrator with the dialogue that follows the long paragraph of which the above cited quotation is a part:
"What's yours?" asked the barman.
"Otro loco mas," said the barman and turned away....