The arena genre is weird. Every 20 something year old gamer I meet claims that arenas were great but no one ever plays them. At the same time we have a snobbishly stringent defacto definition of the genre that no one plays.
We associate Arena Shooters with things like fast movement, "naked" characters that pick up weapon, health and armour, claustrophobic maps or long time to kill. I say, these things needlessly limit the genre:
The necessity of speed makes the skill floor very high, alienating all those who find a similar, more accessible experience in Halo, Teamfortress or even Call of Duty. Yet to me the tactical approach and prevalence of mindgames that make much of the allure of arenas are very prevalent in TF2.
Picking up all gear is extremely limiting for game design. Of course the deathmatch modes and their maps function around it and allow for a very unique experience. However, no one here can tell me that in Capture the Flag or Bombing Run scavenging your base for weapons and ammo before going at it was particularly fun. There is no reason to not allow players to spawn with loadouts. UT2k4 even had its own form of weapon loadouts in the form of undepletable weapon holders next to spawns that gave you a useful bunch of guns and ammo in Assault and Onslaught game modes.
Campgrounds and Ironic are great but did UT2k4 stop being a true arena experience when you played on a huge bombing run map? Why does no one consider the expanded scope of Tribes 2 an arena shooter? It has the mechanics (except loadouts) and story wise it literally takes place in an arena.
Instagib gets rid of item management, weapon diversity and scales down time to kill to an absolute minimum. For many people instagib is an integral part of any arena experience, either as a diversion or as the main attraction. On of the most popular parts of Quake is not even about shooting enemies but about jumping around real fast, so I would even dispute that a good Arena Shooter has to be about shooting.
I consider Unreal Tournament 2004 the pinnacle of the arena experience and one of the best games ever made. It could rival Quake 3 in the core modes and that already made it very good. What set it apart was the millions of different ways you could play this game even without modding. UT2k4 however did not inspire arena developers to make similarly open minded games. Now we are having Xonotic and Warsow, which are fun and offer tremendous amounts of content considering the budgets they have (or don't have) but they are also extremely narrow in vision. On the other end of the budget spectrum the game, the series that has picked up the spirit of UT2k4 is Call of Duty. In Call of Duty you can gear your character in a million different ways and play a dozen game modes.
I am not saying that Tribes or CoD were arena shooters or that we could just get rid of the shooting but when in 2015 a new Unreal Tournament feels exactly like its 11 year old prepredecessor with a few tweaks I'd say our perception of the genre is exactly what made it stagnate. The new UT is good so far and I enjoy it but I don't really understand why I should prefer this over the 2004 version.
To me an arena shooter is probably a shooter about navigating 3D spaces. This differentiates them enough from shooters with (mostly) trivial movement and (mostly) line of sight based combat without limiting them to be strafe jumping deathmatch platforms.