@TheyraAs Zeroth said a few posts/pages back, it's still fairly early in the RP, and as we're still dealing with the "trials" of the Magic Knight selection, now is as good a time as any for new people to join.
As with RoadkilBanana, you're the one who has to decide on what your character's background is - though, as this takes place 150~ years after the events of the canon, and traffic/interaction between the CLover Kingdom and Hino has increased, there isn't strictly a
need to be exiled, unless you want that particular fact as a key motivational point for your character to join the Magic KNights, either to "regain their honor" or something along those lines.
Also, I'm not the GM and have no authority, so take everything I say or suggest with a grain of salt until Zeroth actually replies to you.
When it comes to magic though, I don't think there actually is any sort of
just "summoning magic" in Black Clover. There are ways to create minions or golems via certain elements or magic, but nothing that strictly conjures living creatures or being to serve under your control. The closest thing I can remember doing that was either the guy who could create undead puppets from corpses (and that was his ONLY spell) or the girl who could create a large earth golem from dirt.
If you're going to use Moon Magic, there's only one example of a person in Black Clover using that, so I'd recommend you checking
this wiki page to learn more and see if it's actually something you'd like. Though granted, there isn't much info there, except for the fact that Mono magic is apparently a "special" type of magic, so I imagine it's very rare, like Light or Sun magic.
When it comes to Obsidian magic, I can't really help though, as that doesn't seem to be a type that exists in Black Clover - at least not on the wiki. I imagine it'd be an advanced form of Earth Magic? In any case, depending on what you imagine it capable of, you'd probably be best off running it by our GM.
Hopefully, these answers were use-or-helpful until Zeroth gets back to you with a
real response and answers. ^^