Your GM is correct. Until you are higher level you are not able to cast higher level spells. The entire point of prestige class entry requirements is to gate entry to more experienced characters, and your interpretation basically reduces to just saying "an arcane spellcaster" which if that's what they meant, it's what they'd say. You need to actually be able to cast the spell, so you need appropriate level + high enough stat + correct class + not having swapped out your spellcasting ability for some other ability and so on.
Remember that this is supposed to represent some kind of in game qualification.
"Welcome to the Guild of Master Evokers! So you want to join, eh?"
"Yes!"
"OK, show us your stuff, cast something big - you know, a fireball or
whatever."
"Well, I can't now - but I have the ability to one day!"
"What does this look like, Hogwart's? Don't let the door hit you in
the ass on the way out."
In this context, "able to cast" and "ability to cast" mean the same thing. The designers' neglecting to use the exact same verbiage for every writeup is... pretty typical, actually, and not indicative of any deliberate nuance.
Also there seems to be a misunderstanding about a high stat possibly letting you cast higher level spells early - this is not the case. "In addition to having a high ability score, a spellcaster must be of a high enough class level to be able to cast spells of a given spell level." -d20PFSRD
RAW: Ambiguous
The rules are not clear on what, exactly, constitutes an “arcane spellcasting class” or a “divine spellcasting class,” probably because initially it was obvious. By default, assassins, bards, sorcerers, and wizards were the arcane spellcasting classes, blackguards, clerics, druids, and paladins were the divine spellcasting classes. There was no ambiguity: the former are the classes that cast arcane spells, and the latter are the classes that cast divine spells. Easy.
Then they printed things like Alternate Source Spell, Rainbow Servant, Sha’ir, and Southern Magician, which introduced ambiguity that hadn’t existed before. In these cases, you have spell slots from one class being used for either arcane or divine spells.
Does this new feature suddenly make the class into an “arcane spellcasting class” or “divine spellcasting class” where it wasn’t before? There hadn’t ever been a strict definition before, and they didn’t print one at this point, either. Some of these effects seem to try to include wording that prevents this kind of thing, but much of those rules are also unclear. For instance, consider this from Southern Magician: “The actual source of the spell's power doesn't change,” which Customer Service interpreted as preventing entry to mystic theurge. But it doesn’t really say that, does it? It says something about power source, which is unclear.
Unfortunately, there’s no direct, rules-as-written, “as it says on page xyz of Complete Shenanigans” kind of answer to this question.
Recommendation: Never
RAW is ambiguous, but what’s going to work well in-game is not: never, under any circumstances, should one be allowed to advance wizard spellcasting faster than the wizard does. That should never, ever happen in any game, and if you’re going to allow it you might as well allow Pun-pun.
Allowing these sorts of tricks to qualify for mystic theurge, and other prestige classes and feats that require one type of spellcasting or the other, is pretty clearly legal, RAW, and also usually far less troublesome. The only exception I’d be likely to make is the dweormerkeeper from Complete Divine’s web enhancement, but then I’d probably just ban that class outright.
Even allowing a divine-only prestige class to progress wizard spellcasting is almost-always not a problem. It’s the double-progression that should never, ever happen.
Best Answer
The most recent FAQ entry on the subject confirms that spell-like abilities cannot qualify for requirements, unless the spell name is called out explicitly.
Inner Sea Magic's spellcaster guilds
The Inner Sea Magic has a mechanic for being a member of a Spellcasting Guild, like an academy of magic studies, where you do some quests for them and gain fame within the academy. Then, you can obtain rewards after obtaining a certain amount of Fame, one of which is the Eclectic Training (page 22), which increases your effective caster level on a single class by +1, including spells known and able to cast.
The mechanics, as written, expect a character to gain at least 4 fame per character level, which means that by level 2, you already have +1 caster level from training with an academy of magic. This benefit cannot increase your caster level to higher than your HD, so the benefit is pretty much exclusive to multiclass characters.
Example: A cleric 1/wizard 1 will cast spells either as cleric 2/wizard 1, or cleric 1/wizard 2.
The system also allows you to become member of more than one guild, as long as you can pay for the Entrance Fee, pass the necessary tests and keep up with the Education Exams. The book presents us four guilds:
Kintargo Opera House (Cheliax)
Oenopion Fleshforges (Nex)
Poisoner’s Guild (River Kingdoms)
White Grotto (Absalom)
At 35+ Fame with a spellcasting guild, you also gain another ability called Esoteric Training, which not only increases the previous bonus to +3, but you also gain +1 effective caster level on a second spellcasting class:
The way to increase your Fame score is to pass periodic (between 3 to 6 months) Education Checks, normally 3 to 5 skill checks using a DC of 15 + your ranks on that skill, and pay an additional fee for the new semester. Each check increases your Fame by +1. But you also could do extracurricular jobs and requests for the guild and gain Fame points from it.
Just be careful to not get expelled from your guilds, or the GM could revoke some of the rewards gained. You may quit by notifying your superior and keep your rewards, but you will no longer be able to acquire new rewards.
Note that those academies are not legal on Pathfinder Society, as PFS characters have different guilds they have filiation with, and different boons and rewards that change every season.