The magic of the runes sepparate into 2 subclasses if I may call them so. Rune magic, which is basically words of power, and it enables the user with some unique spells compared to arcane, and arcane, which is the most common one, and it also has some unique spells. Ryze uses both of them, Brand uses only runic cause of the World Rune inside him, Xerath uses arcane and cosmic at once (cause he is an Ascended). Syndra is using spiritual magic, as her spheres were described as being taken straight from the spiriual realm. Basically, runic and arcane are quite similar, cause they originate from the same source even tho sometimes they act differently, and of course have some different spells.
Now spiritual, it also has subclasses. It seems elemental and spiritual are quite conected (ex for elemental: Taliah, Yasuo), and I'm not talking about just fire (which arcane, runic, cosmic etc can also do, tho a bit different each) or frost (which is basically a magic of its own, as frost mages like Yago exist, and Iceborns can use that too, Lissandra is both a frost witch and an Iceborn), because iirc a Rioter stated something about them being conected. If I remember wrong, it's still plausible, because the Freljordian SPIRIT Gods are elemental. Guess who's also elemental? That's right. Elemental magic like Yasuo and Taliah, which seem to be conected to their spirits/souls. Npw for cosmic, it is related to deities and space. Not much to say here except Ascended use that only (Im talking about normal Ascended, not Darkins), except for Xerath who uses both arcane and cosmic at the same time.
Now blood magic is also a thinf of its own. There isn't any explanation needed I think.
There is also shadow (nocturne, zed, kayn) magic and light magic (lux), who are basically opposed to each other, as Lux drove Nocturne back through light magic.
Also, Black Mist magic.
Now to answer the questions regarding the characters you listed:
Veigar: Cosmic, it was described as such in his lore and also through his abilties
Ekko: yes. Hextech tho its not really magic. It uses magical crystals the same way we use electricity.
Twisted Fate: I dont know tbh. I assume he is kinda a weaker mage who needs magical cards to use his powers
The shape-shifters: Hmmmmm... idk haven't read Vayne's lore in like 1 year.
The thorns: Yes. Elemental.
Sylas: A mistery tbh. It seems like its a very special magical ability that only he posseses so we don't know which magic he uses.
The little girl: there are many possibilities: arcane, runic, or elemental. Tho my bets are on arcane cause she made that lighting from herself, not from the surrounding.
Yorick: Black Mist
Hope this helps
