I'm pretty sure deathmatch means the objective is to kill the enemy, instead of capturing areas/flags or anything like that.

This

then how does shootmania fall into this, if they have an objective

Elite mode (3v3) is not standard mode.

I'd be more offended if he referred to Shootmania as an 'arena shooter'. also, I'm too busy being flabbergasted that a game as bad as Shootmania has such a big prize pool to care about semantics.

It's just a sub-label of a much bigger label.

In short, the games you mention are all FPS - I'd say deathmatch is more of a gametype rather than genre;

Portal 2 is a FPS, but not a deathmatch game.
If they added a mod with actual guns and "kill the opponent" objective, the gametype could be called deathmatch.

I'm fairly certain that deathmatch, in its original meaning, is just another term for FFA (hence we have team deathmatch). You could expand on that and say that a "deathmatch" game is a game that emulates the source material for the concept - Doom. Personally, i think Shootmania (CoD and Halo as well, from the list you posted) is too far removed from the tenets of Doom and its derivatives, so i wouldn't consider it being a part of that subset of FPS games.

duel is deathmatch with 2 players then?

Yes.

didn't you notice before?

cause that's what the first post in this thread already said

You could enable weaponstay and remove health and armor pickups and it would still be deathmatch, though.

I disagree. DM to me is an FPS where your objective is to frag everyone (which means you cant have teammates). That is what you do in the "classic" or "standard" game types in Quake (FFA), CoD (FFA) and Halo (FFA). In SM, most game types is FFA, but there are no game type that is the "standard" game type.

Also the only difference from Halo and quake(3/live) (apart from engine, movement systems and different weapons etc) is theme, vehicles and throw-able grenades. Other than that its very much alike, i.e. very much the same genre.

Counter-Strike (There are no FFA) is not a DM game due to teamplay being in focus. Same goes for Team Fortress (There are no FFA) and Battlefield (There are no FFA).

Well, i feel it's less gametype specific and more paradigm specific. Game dynamics are an important aspect of determining what genre does a specific game belong to, and although there is no official canon as far as deathmatch games go, i personally believe only the ones that adhere to the tenets established by Doom (later expanded upon with the Quake series and other spinoffs) should belong there. Hence why i say that Halo isn't a deatchmatch game is because it focuses on different aspects of the FPS experience, same as CoD, CS and other titles that take a different approach.

Still, like i said, there is no canon, so i don't think it's bad in any meaningful way to say that Shootmania is a deathmatch game. It's all arbitrary semantics, really.