Certainly not Oaxaca, as a quick side-note down the central/south of mexico most rural folks don't know how to make cheese, and the aforementioned is one of the few well done ones (and with an interesting presentation) but that's about it, the overrated nature comes from massive marketing, and the commercial version loses plenty of properties of the original (which is pretty expensive for what is worth)
The northern half has the best cheese due to being the rural sector. The most famous one has to be the Chihuahua, which is nothing more than a mild chester with great texture. The best (and real) variant is the one made by its original makers, the mennonites, this one is called Mennonite cheese (Menonita). Tons of flavor, creamy but can go hefty lengths in terms of fusion/melted threads. This is the man's man pick for dinner/strong quesadillas.
Then you have the Asadero variant, which is a fresh and less savory version but makes it up for its use alongside meats and pickled stuff, along with the price. Don't be fooled friend, many rats and peasants will tell you this is the same as Oaxaca, a normal opinion between the ignorant, bloodthirsty class. The elaboration process varies and the milk used here contains much more fats/solids. This is the usual pick for quesadillas at commercial street level.
And the real ones which cannot be flavored normally, the local handmade ones, sometimes called artisan so they can charge you 40% more. You have 2 to 3 variants, the fresco (fresh), the cocido (boiled) and a regional variant, usually called regional depending on the region, which can lead to massive confusion at times, won't tackle those because they are plenty, and usually just copies of european variants.
Going to the point, the cocido ones somewhat vary but at least in the northwest (american southwest) they are similar enough: medium-high fat content, somewhat acid but not that much, cooked in its own whey (which is kinda sour) for around half an hour at 60 degrees, so not really boiled and mildly pasteurized, basic form of presentation are thin discs due to its softness, tortilla-shaped if you like. Very creamy stuff but makes lots of threads, flavor is raw milk with acid tones. This is the usual pick for breakfast quesadillas, but it's also tough to find in non-rural areas and tougher outside the region because there's no preservatives, quantity to ship it well and it's illegal in America due to lack of papers/registration. It's like buying oranges to the guy outside the gas station, except these ones are actually from your city/town, don't rob you, aren't illiterate and aren't escaping from a dark criminal past.
Then the fresco one, there's not a lot to say about this bad boy, or maybe i say this because all latin america has an almost identical variant. This is your standard operation freshly-made usual cheese, can be pasteurized or not, one tastes like salty milk, the other as simply bittersweet raw milk, all of them are moist and squishy. The commercial variant popular in the country (and maybe the only one sold at national scale) is the Panela variant, which is a pasteurized cheese, firmer than usual, tastes like slightly less salty milk. The charm comes from the recipes featuring this as a dresser/accompaniment, the side-product and the mere idea of munching a chunk of soft cheese without feeling bad. Also highly illegal in America due to its informal nature, i heard some injuns in Arizona and New Mexico still make and sell them in the local stores. You can make quesadillas with this but you need to fry it (no, not the deep fry) in sizeable slices, usually served in maize tortilla with strong accompaniments, either northern chorizo or meat. Many caribbeans and central americans love that fry method, they call it simply Queso Frito (Fried Cheese)
The side-product is the funny part, the Requeson, which is basically a sweet but soggier ricotta of subtle flavor in its most usual presentation. While some people might buy a small portion of cottage cheese, the real players buy a kilo of requeson with the same money. A by-product by some producers, this one is cheap due to going foul quickly even if pasteurized, and adding preservatives for packaging makes a mess. Hence why lots of people erroneously don't buy it, this is the main culprit in cases of the formidable Brucellosis, which appears in any unpasteurized product instead of just foul ones as it is in the animal. The sickness itself just fucks your shit up eternally if you eat uncooked products from infected cattle or goats.
There has to be more kinds of cheese, but right now that's on top of my head. TL;DR local chester (Chihuahua, Mennonite), Asadero, Cocido