"WTF difference does it make...."
I doubt if it helps in thinking or looking for such moves but it helps when noting up a game.
I think this;
"....White cannot take the pawn because of the Bishop fork on e4."
Reads and explains better than;
"....White cannot take the pawn because of the Bishop double attack on e4."
I once saw a very lengthy argument on another site about what is a genuine fork.
The argument being, this is a genuine Bishop Fork.
...and this was not a Bishop Fork.
Quite a few people got involved, in the end someone asked what should
they call the second diagram. Nobody could come up with a better term
and that was the end of that one.
I guess the only piece that can claim to have a genuine fork is a pawn
and even then they cannot do it on the 'a' and 'h' files.
This is the bones of one of my OTB games, I was Black. (White to move.)
White played 1.Nd7
I spent a while wondering have I just been Skewered or Knight Forked?