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
A question like this really depends on party make up.
If you're a primary damage dealer, you should go STR, unless you fight primarily big baddies who you can Smite without hesitation (then go CHA).
If you're aiming more towards a tanky person, you should go CON for the HP.
If your party is hurting for heals, then you really want to go CHA for more Lay on Hands.
Based on your description of playing a castigator-type person, I would assume you want to go damage, so I would say STR is probably your best all around choice, CHA if you want more specialized damage, but also more utility in spells/smites/lay on hands/saves.
Best Answer
Check the verbiage about "Temporary Bonuses" and "Permanent Bonuses" under each ability score on d20PFSRD. Temp bonuses give bonuses to specific things that depend on them - skill checks, attack rolls. But permanent bonuses...
Once a bonus becomes permanent, everything else is affected. The most notable example is additional skill points from INT bonuses, bonus spells, etc. "Everything else not specifically listed in the temporary boosts list."
This is a bit of a min-max prevention tool, to stop a stat boost item from getting passed around to, for example, give huge skill point dumps immediately upon someone. It takes 24 hours of "burn-in" to get the more subtle effects beyond adding a plus to a skill/save/attack.