Question on Spell identification

[Ard of El]

According to the text:
"Source Core Rulebook pg. 305 1.1
Sometimes you need to identify a spell, especially if its effects are not obvious right away. If you notice a spell being cast, and you have prepared that spell or have it in your repertoire, you automatically know what the spell is, including the level to which it is heightened.

If you want to identify a spell but don’t have it prepared or in your repertoire, you must spend an action on your turn to attempt to identify it using Recall Knowledge. You typically notice a spell being cast by seeing its visual manifestations or hearing its verbal casting components. Identifying long-lasting spells that are already in place requires using Identify Magic instead of Recall Knowledge because you don’t have the advantage of watching the spell being cast."

My GM says I can't recognize the spell if it is the same spell, which I have slotted as an Arcane version of the spell, because it is being cast from another school such as Occult. (In which I am not trained, so my role would never be better than a failure)

This doesn't seem correct to me. It is the exact same spell! Can someone please confirm the correct interpretation of this rule in this context?

Moved post to here as I had incorrectly posted to general earlier.

[Fuzzy-Wuzzy]

IMO, that it says "...and you have prepared that spell or..." rather than "that spell from that tradition" shows that your GM is wrong. But if they looked at that text already in this dispute and that wasn't obvious to them, I can't find a specific rule that directly addresses the issue. :-(

The text "This entry lists the magical traditions the spell belongs to" for Tradition under Reading Spells on page 306 also has that implication---it's the spell, singular, belonging to multiple traditions, not one spell per tradition---but that's pretty much the same argument....

[thenobledrake]

Your GM is in disagreement with the rule book - whether that's them making a mistake, or them making a house-rule, you're going to have to ask them about.

They may be arriving at this conclusion based on prior-edition knowledge, as previous versions (both of PF and D&D) have had a more strict separation of different traditions of magic - perhaps pointing them to the rules regarding figuring out what spell is on a scroll, and casting a spell from a scroll, will help them see that in PF2 it's the name of the spell, not also the tradition it came from, that matters. Because where back in the day there was such a thing as a "divine scroll" and an "arcane scroll" now there are just "scrolls" so spells of scrolls that show up on multiple spell lists can be used by any of the caster types instead of there being a divine scroll of a spell that shows up on the divine list and the arcane list and an arcane caster can't do anything with it.