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
No, that does not work.
1. What you want to do is rebuilding, which is designed as an exceptional in-game process.
The rules on retraining allow only substitution levels to be retrained. If you want to exchange class levels, this is a rebuild. The rebuild has tough prerequisites that do not allow it to be incorporated into a normal character creation process:
The rules forsee a quest for changing 1/5 of all class levels. Your DM would have to approve your character having completed several quests before entering the game. This is clearly not a regular character creation process.
2. Entering this build by rebuilding would not gain Nar Demonbinder class features.
The point is that "qualifying" for a prestige class means fulfilling its prerequisites before taking it:
Essentially that means you have to meet the prerequisites of the prestige class without taking into account the class features of the class itself. You could not buy skill points with a level of prestige class to enter it. And you cannot take a spell level into account that you only get by virtue of the prestige class itself. These prerequisites (that is why they are called prerequisites) have to be met without the prestige class to qualify for the prestige class. If you change your build so you do not longer meet the prerequisites, you do not qualify for the class.
The example given in the PHB II illustrates this:
This character would still end up with base attack bonus +7, because Dwarven Defender 1 lets you gain base attack bonus +1, but not without the prestige class itself. Thus, if you trade away your wizard levels by rebuilding you no longer qualify for Nar Demonbinder.