Storm Griffons - Storm Gryphons, Hurricane Gryphons, Squall Gryphons, Whirlwind Gryphons, Tempest Gryphons.

I had similar ideas, but "Storm Gryphon" sounds the best. Does the fact that there is a biological species called "storm petrel" somehow interfere with this? And what about vampires, dragons, giants, etc.? However, if people support any of the listed options, I am ready to change it.

"Impulsive" really doesn't sound right. Maybe "Gusty"?

I have no objections.

"Gladiator-Interceptor" sounds normal, similar to "Fighter-Interceptor," and both units have their own functions.

Similar to "Storm Gryphon." If a fighter is named after the fact that it performs the function of interception, that doesn't mean that anyone performing the interception function is a fighter.

Guild Wizard, we need more details about the unit itself.

Ordinary mages with spells, with linear fire, but without direct fire, with an energy channel (this is their main feature). Basic mages.
Other types are called Star Mages (upgrade) and Desert Wizards (alternative).