The screen definitely did the long beep, and as I understand it that implies heart death.
Does that mean she was dead though. Again as I understand it a person isn't technically dead, until oxygen is no longer being supplied to the cells, or something like that. I believe it's called "brain death". It means the person cannot be revived through known means.
On the other hand if Buffy wasn't dead by Buffyverse definition there, then would she have been dead in Prophecy Girl. It would appear the answer would be - no, she would not have been dead in Prophecy Girl either. That wouldn't make sense, so my opinion would be "Yes. By the Buffyverse definition of death, Buffy was dead in Villains."