I’ve heard the term gully-washer way up here in Wisconsin. It’s not used often, but I’ve heard of it.
I tend to use the less colloquial term of “flash-flood” when it’s pouring out. Maybe I’ll use “it’s raining cats and dogs” or “it’s a real soaker.” I guess I tend to focus more on the rain as it’s coming down than what happens to it once it’s hit the ground.
“Gully Washer” is rather common, IME, and I am surprised so many haven’t heard of it.
“Frog Strangler” and “Frog Choker” are a bit more uncommon, but I hear them from time to time.
“Turd floater” seems to be dropping out of favor, I suspect the decline in the use of outhouses–or the popularity of indoor plumbing–is, at least partly, to blame (although gnoitall did mention “log lifter”, which I guess is the same thing).
I grew up in Tennessee, but I did spend over 30 years in Texas.