1) Beastmaster as a limited class. Story quests involve other Beastmasters across Eorzea and other parts of the world. Papaga and Lyon are ones you can meet.
Weapon of choice is like other beastmaster: one-handed axe and shield. You will don in Scouting armor cuz with the introduction of reaper, Scouting set looks lonely.
Your job is to collect beasts, train them up, and pitch them against other Beastmaster in the glory of combat. Aka, pokemon-style, but you fight together with your beasts.
PvP-wise, you have some sort of tournament where you pvp with other players and their beasts.
Group-content wise, you are similar to Blue Mage in the aspect that you can take any role. You can have your pet be tank, heal or DPS, but you cannot change their role in combat. Obviously, they have HP bar, unlike other "stance-users", so pet management is a thing. You have a few skill to protect your pet from the harm away, including teleporting them back to your position. So if you are safe from the mechanic, so does your pet.
2) Minion Master. Can be a DPS ranged class or a healer class, up to what SE has in mind. Limited class is fine too, but it would become something like the Beastmaster above, so I prefer to have them differently.
Weapon of choice is... uh, cane? No. Staff? No. Book? Too many already. Baton? We already have BLU for that. Maybe Ver-rapier, perish the thought. I prefer something blunt, so a one-handed tinker hammer would be cool
Minion Master is another "stance-user", which use Minion to fight (kinda like Lulu's normal attack in FF10). Their minion can be glammed with the appearance of the minions you have.
Playstyle-wise, unlike Summoner, Minion Master melee to build gauge, and use the gauge for the minion to execute different attacks that build another gauge where the master can use to deal even more damage with his hammer. Or whatever floats SE's boat. My idea is kinda dry.