The reason for asking this is there is an ongoing PR on the Lichess Opening database repository which needs clarification.
To recap some history from the Wikipedia page on the Blackmar-Diemer Gambit:
Armand Blackmar was an American 19th century player who played the gambit 1.d4 d5 2.e4 dxe4 3.f3, which was referred to later as the "Blackmar Gambit". It's not very good as black has 3...e5!.
Then later Von Popiel introduced 3.Nc3, with the idea 3.Nc3 Nf6 4.Bg5, and analyzing 3.Nc3 e5 which was called the "Lemberger Counter-Gambit".
Then Diemer took both ideas and introduced 3.Nc3 Nf6 4.f3, when 4...e5 isn't very good as 5.dxe5 hits the knight. That's the Blackmar-Diemer Gambit.
So what is the answer to "which one is more reliable (or correct)"?
In my opinion, strictly speaking 1.d4 d5 2.e4 does not have a name. The name depends on what happens after.
In practical use I guess chess players would call it Blackmar-Diemer Gambit, but that goes hand in hand with the fact that it would be expected to continue 1.d4 d5 2.e4 dxe4 3.Nc3 Nf6 4.f3 if it's played at all these days.
But if you want names to be correct in a way, then I think it would be best not to give the position after 1.d5 d5 2.e4 a name . After all, Black can just as well answer 2...c6 (Caro-Kann) or 2...e6 (French), and the game would at no point have been a Blackmar or Blackmar-Diemer gambit.