Lighter and crispier, huh
It sounds like you want more of a fried style exterior, not a toasted one, so using a layer of oil or liquid grease in the pan is probably a good move. A thin smooth layer in the pan will give the tortilla surface a "deep fried" sort of texture (the tortilla only cares if its surface is coated in enough oil, not whether there's extra) - and the extra heat the oil conducts should crisp it up faster, giving it less time to dry out when cooking - the lower and slower method works if you want a softer crust. Remember to move your tortilla around when first putting it into the pan, so it doesn't stick (instead of trying to un-stick it later) - that technique is still useful even when cooking with more oil.
Along those lines, it may be wiser to pre-assemble your quesadillas, possibly separating your cheese into two thin layers (over and under) to encourage quick melting, and settle the whole thing in the pan at the same time instead of layering on once on the heat. Since you're using hot oil, you should end up with a shorter cooking time (a tortilla laid in hot oil can start crisping in a few seconds) - that means less time for your ingredients to heat up inside the tortilla. If your ingredients are room temperature or possibly microwave-zapped if something takes longer to heat, anything is pre-cooked that needs to be, and you only have it in the pan long enough to melt the cheese - you will find it much easier than otherwise.
Alternatively, if you're only making one or two, you can cook the first side crisp, turn the tortilla over (tricky if the cheese hasn't melted yet, but doable with a spatula and plate, or plate and hand), and turn off the heat. The second side will crisp in the initial residual heat, and a minor drizzle of oil around the edges will give it that oily texture if there wasn't enough, but the heat will quickly drop to let the interior warm through and the cheese finish melting without overcooking the exterior...much. It takes practice to get it right, but it works.