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.
That... doesn't really work like that.
Heighten Spell is unique in which it actually changes the effective spell level:
Unlike other metamagic feats, Heighten Spell actually increases the
effective level of the spell that it modifies.
It doesn't add levels to a spell, it changes the spell level. That's subtly different.
A Silent Fireball is a 3rd level spell, even if it uses a 4th level slot. That's a +1 level increase.
A 4th Level Fireball is a 4th level spell. That's a +0 level increase.
Magical Lineage can't reduce the effective spell level, just the extra levels added by metamagic feats. Since it can't enable you to cast a 3rd level spell as a second level spell, it would make sense that heighten spell would not work with magical lineage.
Best Answer
This argument comes up from time to time, but it doesn’t really stand up to scrutiny. For example, a 4th-level sorcerer only knows one 2nd-level spell; do they therefore fail to qualify for mystic theurge? A 3rd-level wizard might be able to cast only one 2nd-level spell per day; do they fail to qualify for mystic theurge?
Really, though, the rule is ambiguous enough that you can’t really refute this argument. Someone can argue that the sorcerer qualifies by dint of being able to cast that spell more than once a day, while a wizard qualifies by having multiple options he could prepare: these are different kinds of plurals, but one could still argue that each is a plural, and the Precocious Apprentice is not. The rules do not actually expand on what it means to be able to cast “spells” of a given level.1 Precocious Apprentice certainly isn’t addressed specifically anywhere.2
Anyway, if this argument is enforced as a rule, you can still work around it. A forcused specialist wizard (Complete Mage) swaps one spell slot for two spell slots that must be used to cast spells of their specialist school: RAW, that will apply to the Precocious Apprentice slot, and the two slots you get out of it won’t inherit the restriction on Precocious Apprentice’s slot. At that point, the 1st-level wizard can cast two different 2nd-level spells in a given day, if he likes, and there is no argument I can think of that this would fail to meet the requirements of mystic theurge et al.
Ultimately, I think a DM should be more honest about this kind of thing: don’t get into a RAW argument, that’s irrelevant, just make a ruling and/or houserule. The DM isn’t constrained by what the rules say, regardless of how you interpret it. The DM is free to say the mystic theurge has no prerequisites at all if they want, or that it requires 3 levels specifically in cleric, or whatever else, and that’s legit as far as the game is concerned.3 Trying to get picky about a plural or not is just asking for trouble, because the rules simply aren’t written to that level of pedantry and a DM will set an unfortunate precedent that way. Better to handle Precocious Apprentice itself directly with a houserule than try to argue that the trick doesn’t work.
And really, that’s how all such RAW abuses should be handled. People get up in arms about RAW corner-cases, and some get really caught up in arguing that they don’t actually work, and most of the time it’s completely irrelevant. If you ask your DM to enter mystic theurge as a 1st-level archivist/1st-level wizard with Precocious Apprentice and Southern Magician, the question shouldn’t be “does it matter that mystic theurge uses a plural?” or “what does Southern Magician mean when it talks about power source?” or “should we pay any attention to CustServ?” Those questions can be relevant, ish, when discussing things in an abstract way, particularly in a theoretical-optimization project, but you have a DM there to decide things. The question for the DM isn’t “what is the absolute RAW way to run things?” the question for the DM is “what ruling is going to increase the game’s fun the most?”
A 1st-level archivist/1st-level wizard/mystic theurge might be overpowered for your game, and bad for it: then don’t let it happen. A 3rd-level archivist/3rd-level wizard/mystic theurge might be (almost certainly will be) underpowered: then don’t put that in your game either. Figure out where the balance lands that produces the most fun, and go with that—and don’t pay the least attention to what the rules say or what their authors meant or any other irrelevant details. Those are questions for another context, not the one where you’re trying to decide how to run your game.
Aside from the discussion in Complete Arcane of the warlock qualifying for prestige classes, but that’s only discussing how spell-like abilities like the warlock’s invocations don’t qualify; it’s not relevant to the discussion here.
The FAQ actually does—and despite the problems with the FAQ in general, it’d still arguably be better than nothing here—but unfortunately that mention doesn’t make a whole lot of sense: “In the Sage’s opinion, the Precocious Apprentice feat would not help you qualify for a prestige class or feat because it gives you a chance at casting a 2nd-level spell, not the inherent ability to cast 2nd-level spells,” (emphasis original). By that logic, a wizard wearing a chain shirt—with its 20% chance of arcane spell failure—would not qualify for prestige classes either, which hopefully we can all agree would be nonsense. The Sage is, like your DM, attempting to pretend a rules abuse doesn’t exist by torturing the wording to mean what he thinks it ought to. I cannot more strongly recommend against playing that game, as my answer discusses at length. If you don’t like Precocious Apprentice qualifying for things—an entirely reasonable position—just rule against it. Don’t try to pretend your ruling exists somewhere “between the lines” or whatever.
I strongly recommend against either of those houserules, however.