Google seems to recognize the third possiblity, "puff". I would definitely avoid the word "poof".
Pouf is quite common. Never heard of "lather builder" except for the possible case that the company selling it was foreign and did not know the word pouf.
I don't doubt it's correct, since Rover has seen it, but I don't see the expression in The Free Dictionary.I'd never heard of "lather builder" either though, but it's correct according to Dictionary, Encyclopedia and Thesaurus - The Free Dictionary
My daughter bought one this very day in Florida.
It's called a lather builder.
Esthetic Supply Company // Products
Rover
Is there any difference between how poof and pouf are voiced?
As far as I know poof is a rude synonym for gay.
So I guess a request "Give me a poof" may be treated in ambiguous way, isn't it? :-D