Openings are not about move order. I don't play questionable gambits like the Blackmar Diemer Gambit or the Hebden Gambit, so I am not 100% sure what the difference is. I want to say it has to do with the Nc3 move.
That said, the opening of the game is based on the position, not the specific sequence of moves.
For example, after the move sequence 1.e4 e6 2.d4 d5 3.Nc3 Nf6 4.Bg5 Bb4, you have the MacCutcheon French. The MacCutcheon French is the position that results from those 4 moves, NOT the moves themselves.
This is where "Transpostions" come into play.
1.d4 d5 2.Nc3 Nf6 3.Bg5 is traditionally called the "Veresov" Attack, and if Black plays something like 3...c5 or 3...Nbd7, that's what you have, a Veresov.
However, let's say Black plays 3...e6. If White plays something like 4.e3 or 4.Qd3, you still have a Veresov, but if White plays 4.e4, you have a direct transpostion to the French Defense, and if Black plays 4...Bb4, you have a French MacCutcheon.
While it may have started out as a Veresov, it no longer is a Veresov, it's a MacCutcheon French. Doesn't matter whether it's via 1.d4 d5 2.Nc3 Nf6 3.Bg5 e6 4.e4 Bb4 or 1.e4 e6 2.d4 d5 3.Nc3 Nf6 4.Bg5 Bb4.
So therefore, to answer your question, after 1.d4 d5 2.Nc3 Nf6 3.e4, what opening you are actually playing depends on Black's response. If Black plays 3...e6, you've directly transposed to the French. If he takes on e4, you've transposed to either the Blackmar-Diemer Gambit or the Hebden Gambit.
I was wondering if White is to play a true Blackmar Diemer Gambit, does White have to play the move order 1. d4 1...d5 2. e4 2...de4 3. Nc3
If, for example, white plays 1. d4 1...d5 2. Nc3 ? 2...Nf6 3. e4 If white plays 2. Nc3 2...NF6 3. e4 3... Can the move order (2.Nc3) still be called a Blackmar Diemer Gambit?