My friend and I were talking about TES lore and how TESO got some (...) things wrong, and the College of Winterhold and how it pertains to the Mages Guild came up. And now we're a bit confused.

According to sources, Shalidor ("dubiously", admittedly) founded the College of Winterhold in the First Era. The Mages Guild was founded by Vanus Galerian early in the Second Era. The Mages Guild and College of Winterhold disagree on Necromancy, and remain separate institutions due to this and other (?) arcane disagreements.

However: Shalidor, who, according to sources, seems to be the same Shalidor as the one who founded the College of Winterhold, is seen as the Arch-Mage of the Mages Guild in The Elder Scrolls Online. According to what I've read, TESO lore is to be assumed just as valid canon as that of the previous, single-player games (but correct me if I'm wrong).

Why would someone who founded an essentially rival institution end up as the Arch-Mage of the newer Mages Guild? I can really only think of three explanations:

1. The College of Winterhold was, at some point in its history, part of the Mages Guild.

This isn't backed up by the Mages Guild apparently being a younger institution than the College of Winterhold, and they seem to have been in disagreement all along, making it unlikely that a) the College of Winterhold split from the Mages Guild at some point or b) the College of Winterhold and the Mages Guild decided to form a coalition at some point.

2. The Mages Guild was originally part of the College of Winterhold.

I should think there would be a source mentioning this if it was true, and I wasn't able to find any based on my (admittedly quite quick and non-thorough) Mages Guild research pertaining to this particular subject, nor have I ever heard this before. Everything just mentiones Vanus Galerian founding the Mages Guild, end of story.

3. The Mages Guild's Arch-Mage Shalidor is different from the College of Winterhold's founder Shalidor.

This is not backed up by any evidence. These Shalidors both would have lived in approximately the same time and same place, and both would have been exceptionally powerful wizards. It's unlikely that this would happen.

I suppose it's also possible that Shalidor turned his back on the College of Winterhold and joined the Mages Guild, but it seems quite unlikely to me, especially since he'd have been going from an institution he founded as one that would encompass all types of magical learning to join one founded later on and with more restrictions. Perhaps Shalidor had a change of heart regarding necromancy, and jumped ship?

Has there been some official information on this that I've missed? Does someone here know better? I'm personally not very versed in Mages Guild or Skyrim lore, since neither of those are included in the parts of TES lore that particularly interest me, but this is really bugging me now!

EDIT #1: After looking up more on the subject, apparently Vanus Galerian is still alive and kicking in the Elder Scrolls Online as well, but only involved in some of the very high-level quests and very few of them, at that. He's titled Archmagister, which would rank him below Shalidor... in a guild he himself has founded? I suppose he could not want all the responsibility of being Arch-Mage and would want to retire or something, but that too seems a bit odd.