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Looks like you have two different types there. The caps in the first two pics are striate and look like common mycena , the second ones have me stumped, maybe some kind of entoloma or psathyrella but i don't think so. With little brown mushrooms spore prints are usually necessary to get even to the genus.
"P." is not another way of saying Psilocybe. It's an abbreviation that has meaning only in context after a genus name has already been mentioned, and "P." simply refers to ANY genus name already mentioned that begins with a "P". If multiple genera beginning with the same letter are mentioned, then you use the second letter as well, to distinguish them.