It does not surprise me at all that the vet didn't know what it was. If you go to your doctor with a growth he will be unlikely to tell you what it is, but send you for a biopsy.
A budgie can't tell the vet where or how much it hurts so a vet really hasn't got a change. A biopsy for a budgie is prohibitively expensive. If you tell the vet you want a diagnostic test done they will do it, providing you pay up front.
It could be several things but seeing another growth developed recently it is obviously not on the mend.
I don't like to be the bearer of sad tidings, but from what you've written, I am afraid things don't look too good. Sorry.
All I can suggest is to keep cleaning it and hope for an miracle.