What about Nature Magic (Ivern, Maokai, Zyra)? Related to spiritual magic?
Not sure about Ivern, but Maokai is a nature spirit, so obviously its spiritual. Zyra is kinda of a mix, there was elemental magic in the soil she came from but it was mixed with a burst of runic magic used by a human sorceress that later gave birth to Zyra.
Shapeshifting (Nidalee)? I know Udyr's is linked to spirits (I.E, spiritual magic), but I don't know if Nidalee works the same way (considering she physically changes into the animal).
Nidalle is confirmed to have diluted Vastayan blood(as is Udyr) and the Vastayans are deeply linked to spiritual magic. Waaaarghbobo explained it better in this comment:
People with vastayan heritage retain some of this power but have much more limited control. They might be able to transform quickly-- but only to one or two shapes (usually related to their spirit animal). Or they might be able to "partually" transform to many shapes or harness natural magic...but they aren't really in control or can't totally transform.
There is also a guy who can transform into a pig in Santagelo Vastayan Journal and explained that he could transform because of his diluted Vastayan blood.
Shadow magic (Nocturne, Kayn, Zed)? Linked to Void magic?
Shadow magic is more linked to spiritual magic, it is a form to force your will over spiritual magic and corrupt it, it was better explained in Nocturne's lore:
Although no record names the first of them to cast off their flesh and enter the spirit realm, it is known that they came to stalk one another not only on the battlefield, but in landscapes shaped by their own subconscious thoughts and emotions. Unconstrained by the laws of physical reality, they fought in ways that more mundane minds could scarcely comprehend, even conjuring subtle, etheric assassins to do their bidding. Shadow mages seemed particularly skilled at such things—and so it was, for a time, that they came to dominate the spirit realm, casting it into twilight.
And anyway, Zed short bio call it spirit magic:
During the war, desperation led him to unlock the secret shadow form—a malevolent spirit magic as dangerous and corrupting as it is powerful.
Sand magic (Ascended)? Some form of Elemental magic?
The Ascended uses celestial magic, but it is deeply shaped by their own percerptions of the concept they fused with and the magic itself.
For example, Azir most likely fused with the concept of leadership, since he is the emperor of a desert-like place, when he received powers associated with the leadership, he received powers over the most abundant resource of the land he leads, sand. Same explanation applies to his soldiers.
But not talking spefically about Ascended, Waaaaaarghbobo said that sand can be considered a "core element" depending on the local culture and that would lead sand magic to be type of elemental magic in a certain place, in Shurima it is really likely that some people have elemental sand magic.
Poison magic (Cassiopeia)?
Wouldnt say there is such thing as poison magic, Cassiopeia was simply transformed into a half snake and received "poison powers". The poison was probally created by an Ascended(Naganeka?), my current guess is that the venom was meant to transform Cassiopeia into a snake like beast that would devour anyone who came to the tomb with her, but the venom was weakened, either for the passage of time or some unknwon reason and turned her into a half snake instead.
But back to the main matter, if the venom was made by an Ascended, it is probally some type of celestial magic, which is, again, heavily shaped by perception, that Ascended probally represented the concept they merged with as a snake(animals are often used to represent concepts in Shurima) and received snake like powers, including magical poison, because of that.
Spider magic (Elise)?
Elise received her new form from Vilemaw's poison, I believe that Vilemaw is a spirit-god(like Ornn, Anivia etc) which was corrupted by the Black Mist in some way. That would means that her magic is spiritual.
Holy magic (Kayle)? Related to Light magic?
Dont really think holy magic is a thing, Kayle is probally going to use celestial magic but until the rework I cant say anything for certain.
Illusionary magic (LeBlanc)?
I think thats just a more advanced type of light magic.
Lee Sin?
Lee Sin canalizes the power of the Spirit of the Dragon, therefore his magic is spiritual.
Faerie magic (Lulu)?
Pix was mentioned to be a spirit in the last ask Riot:
Lulu has taken this to mean that he was born as a seed-pod, and later sprouted into the fae spirit he is now.
And Yordles are also spiritual in nature, since their home, Bandle City, is a place of the spirit realm, so yeah, spirit magic.
Dark magic (Syndra)? Related to Shadow magic?
Syndra seems to be a reeeeeeeeeeeeeeeally unique case, I'm not sure what magic she has but I dont think it is related to Shadow magic. I could enter in depth but...some persons made me tired of dicussing Syndra :^).
Those are the ones I can answer, hope that helped.
But I want to leave something clear, while all those mentions of spirit and celestial magic might feel limitating to a certain extent, they are only a reference to the realms they originate, spirit magic include things from Zed to Neeko while celestial magic includes things from Aurelion to Renekton. As you can see their origins in no way means they are equal to all other subtypes of their respective magic.