I'm no expert on southern climate grasses, but shade could be part of your problem. How much direct sun does the lawn get each day? Does the house shadow cover the entire area, etc? Bermuda requires full sun all the time. St. Augustine does better in shade, but still requires a good 5 hours a day of direct sun. Have you fertilized, and if so how?

How much rain do you get? Does the lawn puddle up and stay wet for long periods, or does it drain quickly? Ideally you'd get a soil sample done, if you really want to do things right, but I don't know where you'd get a good test down there. Is it expensive to ship items to the US? Surely, though, there must be someplace down there where you can get a decent soil test.

Digging up the soil is a last resort as it isn't usually a good thing to do. Most soils can be fixed, if in fact anything at all needs to be done.

If you want a better ID on the grass that you have, try to take a picture in the shade or when it is cloudy. Get the camera as close as you can, 3 or 4 inches, from the grass and also low to the ground. Take a couple of photos and post them here.

Can you show some GOOD closeups of the grass plants for identification"

But I am guessing I would have to fix the soil. What kind of soul do I
buy? Should I just dig up everything that is there and start fresh?

You fix the soil by treating it well, not replacing it.

After we have identified the grass, we can figure out what sort of mowing and watering schedule it needs, and what kind of fertilizer, to get it looking better.

No need to apologize!

With the shade in the pictures I don't think you will be successful with anything but St Augustine. It will grow great in that area. If you can find even a small piece you can plant it and it will spread to fill the entire lawn.

I would start to improve the soil by putting down organic fertilizer. What I'm talking about is something like alfalfa pellets (rabbit food), corn meal, or whatever ground up nut, bean, or seed you can find at a farm supply store. It needs to be ground up. That's the only real requirement. Apply at a rate of 15 pounds per 1,000 square feet (7-10 kilos per 100 square meters). Whatever organics you find can be used as your only fertilizer from now on or you can use chemical fertilizers. The organics will make the most improvement for the first application. If you use that now you might not even want to change the grass, because it will help that much. Give it 3 weeks to go to work and then reevaluate.

Watering: Deep and infrequent is the mantra for watering. This is for all turf grass all over the place. Deep means 1 inch all at one time. Put some cat food or tuna cans around the yard, and time how long it takes your sprinkler(s) to fill all the cans. Memorize that time. That will be the time you water from now on. My hose, sprinkler and water pressure takes 8 full hours to fill the cans. Your time will likely be less. I like gentle watering. As for watering frequency, that depends on the daytime air temperature. With temps in the 90s, deep water once per week. With temps in the 80s, deep water once every 2 weeks. With temps in the 70s, deep water once every 3 weeks. With temps below 70, deep water once a month. Note that you have to keep up with quickly changing temps in the spring and fall. This deep and infrequent schedule works in Phoenix and in Vermont, so it should work for you. The reason for deep and infrequent is to grow deeper, more drought resistant roots and to allow the soil to dry completely at the surface for several days before watering again. If it rains, reset your calendar to account for the rainfall.

Good job on the photos. I'm sure a warm season grass expert will be along. The other thing I noticed about your plot is that it seems to be way below the grade of the sidewalk. Does the entire roof drain down onto the lawn when it rains?

If fertilizer doesn't do anything, then definitely put something organic on it. I would start with corn meal, which should be available easily either at a feed store or at a grocery store as corn flour or masa de maiz. You want plain corn flour - not the stuff that is tortilla premix.

The grass looks like centipede. Centipede thrives on no care at all. I have seen rare cases where it looked good, but usually it looks bad. That could be because I hang out on these forums where people only write in with problem lawns.

Fertilize with organics and give it 3 weeks. Then we can reevaluate everything.

If you don't have a small hand spreader, you can simply broadcast it out by hand like feeding chickens. No need to poke holes.