I'd say a summon could be whatever you want. It could give you a power up/transformation, a weapon, a mount, or show up and attack directly itself. 

Like for example, Odin could grant someone his horse and sword, or Bahamut could rain swords from the sky without appearing directly (There's so many versions of Bahamut at this point...). Someone like Shiva is more of a direct, appear and act on her own, same with the Knights of the Round. They are summoned to attack as their own entity. But summons can work however you want once your character gains their power. And doesn't need to be bound to one way. 

You could summon Odin's horse in one fight, and have Odin himself appear in the next. (Unless it's FF13 where... Odin IS the horse...)

Probably the main thing to think about is what each summon stands for. Like, for obvious reasons, I don't see anyone but my character being able to summon the Knights of the Round. 

Someone like Titan is the embodiment of the sheer strength of the Earth itself. Garuda can embody wind and freedom. Ifrit... kinda tends to represent the fires of hatred and anger. 

And yeah, it should at the very least take a lot more to earn the power of a higher summon. Like, gaining the more baser ones like Ifrit, Titan, and Garuda, would be a walk in the park compared to earning the power of Bahamut, Ultima, or Omega. And then there's just a wide range of power between them from something like the Moogle King up to Alexander.