My own thoughts: Given that they're more equality types, it could be that they see the war as a collective, cooperative effort between all of them as individuals, and rather than orders, they simply try to assist each other and organize plans together rather than some kind of subordinate/superior system.
I like this line of thinking, however a ranking system may exist solely based on the "skill"/"strength" of a persons psionic abilities. Like, someone who can simply pick up surface thoughts would be not as powerful as a person who could search an entire room of people for a specific piece of information.
For a more in depth thing regarding how they might fight though, I imagine the mage's would have Golems and other magical creatures/summons/creations as their "heavy units" while mages slinging spells would be their basic infantry. The steampunks probably use all sorts of weird technology and weapons, a lot of it probably air-oriented. The Psionics, I would think, would have their own psychic-tech and just plain psychic powers, so probably a lot of telekinetic stuff going on - A thing they might focus on is using their powers to be amazing at long range, having scouts and such relay precise information telepathically to a submerged weapon out in the ocean, or to help guide long-ranged weaponry towards a target.