I’ll make it short.
Pus is the liquid formed in infected tissue
Puss is a term of affection for a cat, or a coquettish (young) woman.
Pussy (pronounced to rhyme with “fussy”) means full of the above liquid. Pussy (calm down … and pronounced with a u as in “pushy”) is another affectionate term for a cat. Maybe that’s how the confusion has crept in.
Anyway, there we go: all done now. Eugh. But necessary, yes?