First, a note: It is entirely within the DM's purview to decide that a sorcerer is unable to do this, and/or that a wizard is able to, as wish states that (paraphrased): "If it's not in the list, the DM has the final say." However...
Your DM's decision appears, at first glance, to be based on faulty reasoning.
According to what you said, the crux of the matter is that your DM believes "he wouldn't be intelligent enough to phrase it in that way". Considering that you said your DM would allow a wizard to phrase it that way, this suggests that your DM believes that sorcerers are automatically less intelligent than wizards, and thus inferior to wizards at anything that depends on intelligence. If I would hazard a guess, this is based both on the two classes' spellcasting stats (Int for wizards, Cha for sorcerers), and on the stereotype that wizards are studious scholars of magic, but sorcerers use natural talent as a shortcut.
This, however, isn't supported by the game itself. The spell wish doesn't contain any references to intelligence, nor to your spellcasting ability. This means that, as written, its efficacy isn't based on how you use and understand magic, but on your competence at doing so (i.e., your level in a class with full spellcasting progression, not your stats).
Furthermore, the mechanical descriptions of the classes in question don't provide anything that could answer this question; mechanically speaking, a sorcerer has a smaller spell pool, but is better at using and modifying the spells they know on the fly (Metamagic lets them adjust their spells as they're being cast), and may have an instinctual understanding of how magic works (Flexible Casting allows them to reshape their magic reserves on the fly). This would suggest that both are (on average) equally intelligent when it comes to casting, with the main difference being how they view magic: wizards approach it academically, studying it and quantifying it as a known science so they can use it by rote; sorcerers approach it empirically, testing its boundaries and experimenting with their spells' capabilities to see what they're capable of. From this train of thought, the sorcerer can be considered to be the street-smart counterpart to the more book-smart wizard; both have the same potentional for intelligence, with the main difference being whether their knowledge is based on depth or breadth.
Furthermore, the flavour text for sorcerers states that they're "every bit as useful to an adventuring party as wizards, making up for a comparative lack of breadth in their magical knowledge with enormous flexibility in using the spells they know" (PHB, pg.100). This suggests that their spells aren't any less potent than a wizard casting the same spell, and in fact could be interpreted to support the sorcerer understanding the few spells they know better than a wizard does. This, of course, is mere speculation, due to relying on flavour.
The most convincing evidence, though, is the use of the term "spellcasting ability" throughout 5e's magic mechanics. The rules of the game don't distinguish between what stat you use to cast; rather, they state that all means of casting a spell are in fact equivalent (or at least, all abilities are equivalent when it comes to casting spells). By the rules of the game, a Lv.10 wizard casting, e.g., hold monster with 18 Int will be exactly as effective as a Lv.10 sorcerer casting hold monster with 18 Cha (both have a spell save DC of 16).
We can thus be reasonably certain, due to both this and the lack of evidence to the contrary, that this decision is not strictly based on RAW.
This doesn't necessarily mean that their decision is wrong.
While there's no reason to believe that all sorcerers are less intelligent than all wizards, they may be thinking of your character specifically. If, say, your sorcerer has an Int of 8, and the hypothetical wizard has an Int of 16, then it may be a reasonable assumption that they would be unable to locate and close any and all loopholes that could prevent their wish from being fulfilled as intended (and that the wizard could). This, of course, is less likely if your sorcerer does in fact have a high Int score, but I have to mention it on the grounds that it might be part of the reasoning.
[Note that I'm not saying I think your sorcerer is stupid. I'm just trying to explore all potential sources of their claim, since you didn't mention your sorcerer's Int score.]
This, however, is where things become subjective. If their logic is based on the classes in general, then we have a few rules to back the "they're equal" mindset up. If the logic is based on individuals, however, then all bets are off.
Alternatively, they may have decided that this isn't actually a question of intelligence, but of knowledge. Perhaps it isn't common knowledge in this setting that wish has the potential to function as a literal (or possibly even malicious) genie; maybe this particular information is only recorded in certain dusty, old tomes, stored in a library that only wizards have access to. If this is the case, then it's entirely possible that your sorcerer wouldn't know that they needed to close any loopholes, and thus that their intellect isn't actually the main factor.
Furthermore, it appears that their decision is based on the legalese, and not on the spell itself. It thus stands to reason that the DM has no problem with wish being cast in that manner (as they said, they'd let a wizard do it); their decision is based on the belief that sorcerers lack the intelligence to provide such wording.
I would even go so far to suggest that you ask them whether their answer would be the same if faced with a 15 Int wizard and a 20 Int sorcerer, where the sorcerer is in fact provably mechanically more intelligent than the wizard. Don't be confrontational, just treat it like a point of curiosity based on their initial decision.
You do have other options, however.
You said it yourself.
My Dungeon master would let a wizard cast it in that way though.
This statement has two components, one of which can be very helpful to you.
- Your DM would allow a wizard (or perhaps any sufficiently intelligent character?) to provide sufficient legalese to close any potential loopholes in a wish.
- Your DM would allow such legalese to be used when casting wish.
Logically, if your character is unable to provide the legalese themself, shouldn't they be able to ask the wizard to word the wish for them? This would allow the wizard to meet the intelligence requirement for you, which should in theory comply with their statement while still allowing you to make your wish yourself.
If they say no to this, still claiming that your sorcerer wouldn't be intelligent enough even when another character is explicitly handling the part that would require intelligence, then it may be worth having a more in-depth discussion with them, to determine whether they intend for sorcerers to be weaker casters than wizards.