This would lead me to lean, for an Arcane-powered warrior, to their central/base archetype to be dependent on having and using power to defeat enemies that are supernatural, defending the physcial world from the supernatural, on supernatural terms with knowledge/training/experience in supernatural means. This could be "witch-hunter/the Witcher" types of specially trained "hunters." Could be "knightly" orders of Jedi-esque Elves keeping a magically eye on the functioning and safety of sorcery and sorcerous/otherplanar threats upon the world. Could be someone trained out of personal motivations (or plain greed) to find and master various arcane items and creatures...purely for the increase of their own power/ends (akin to the iconic Magus from Pathfinder). All of these characters could be the arcane-warrior. Some "feel/look" like a paladin with arcane magic. Some feel/look like a ranger with arcane magic. Some look like "Bladesingers" or "Magi" or "Duskblades" or <insert preferred specific name here>. But they are all just different flavors of the spell-wielding weapon-trained combatant....the name/title of this class is what the real issue is. Not its 'identity," per se.
"Swordmage" is so "blah" generic. This also applies to all of the "just put two words together" nonsense: "Spellsword, Duskblade, Hexblade,"...even "Bladesinger," etc... Besides several of those are too specific in flavor/story to be a base archetype class name.
"Magus" is kinda taken.
"Gish" is just made up non-word nonsense horrible that should only ever be used in reference to githyanki...if at all.
"Guardian?" I like! But it does, rather, put a stranglehold on what the explicit presumption of this character is to be. "I don't want to be a Guardian! I want to be a magical marauder!" Now, if you were very clear that the class name was in reference to a character who is looking to "safeguard" magic/the supernatural to any cause: from keeping arcane magic and creatures in the world to eradicating it entirely [so you are the only one left with arcane knowledge and power] are all plausible for someone calling themselves a "Guardian." I guess it could work.
My own version of this class is called a Sentinel. Rangers range. Sentinels "keep watch." They are alert and paying attention [to magical things] and "watching"...but are they watching to defend magic in the world? Sure. Alert to magical goings-on for their own purposes/selfish ends? Yup, that too. Paying attention/learning about magic to stop its encroachment or possible destruction of the world? Could be that too. Are they the "sacred" order of magical [arcane] archers from the high-elf kingdom responsible for the direct protection of the elfin sorcerers council? Sure are! Are they medium armor-wearing "battlemages" -more interested in flinging spells than swordplay, but they still carry/know what to do with a sword if needed- from the nation of the Archmagus Imperialis? Yup, them too.
So, the concept/identity is simply: a weapon-trained combat-capable (melee and/or ranged!) warrior who knows arcane spells, possesses arcane knowledge, and expertise encountering/dealing with/defeating "arcane creatures" and magical threats.
The problem is that the Fighter/Mage -from D&D's incarnation- has never HAD its own base class. It doesn't have a "name."
Say "Ranger" and all D&D (and any fantasy RPGers, computer and table) know what/who you're talking about. NOW, those imaginings can be wildly different depending on one's age, game, style preferences, all kinds of things. But everyone will have some image/idea, automatically, of what "Ranger" means. Same with Paladin. Same with Bard, and so on. For "Fighter/Mage" character concept...we don't and have never had a convenient 'Label" like that.
Basically, the solution is, D&D developers need to PICK something and just stick to it. Just use it over and over and over until it is just an assumed part of D&D/fantasy game-play. ...but, preferably, not something "hokey" like "Spellsword" or "Swordmage."