It's hard to say with those pictures, but St Augustine is certainly a possibility. The tell tale sign of St Augustine is the blunt tip on new growth. Other grasses come to a point, but not St Aug.

Assuming it is St Augustine, here's how to care for it.

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.

Mulch mow at your mower's highest setting all the time. There is never a good reason to lower the mowing height for St Augustine. You can mow it every week or let it go a couple weeks.

Fertilize no earlier than Memorial Day. Then again on Labor Day and Thanksgiving. Having said that, in your case I'm going to suggest going on an all organic program for at least this year. I would suggest corn meal if you can find it. Apply at a rate of 20 pounds per 1,000 square feet. You should be able to find it in 50 pound bags at your local feed stores, but definitely call ahead first to see if they have it. Cost should be $10 or less. I'm not talking about corn gluten meal. That product is more like $40 per bag. Apply corn meal every month or every other month as your budget allows. You can also use corn flour from the grocery store. It comes in 25 pound bags and costs about $10, too. Look on the aisle catering to Hispanic shoppers. Don't get the one with baking soda and baking powder in it. The bags look alike, so read the ingredients.

In picture 2 it appears the soil level is below the concrete by at least an inch. That is the reason the grass looks so poor right there. The soil should be brought up to the same grade as the concrete. If you cover it all in one application, it will kill the grass underneath. The surrounding grass will grow back to fill over it, but it will not look great until then. You could put some sand or topsoil in there every week to build it up slowly.

There is no seed for St Augustine. It grows like a carpet and will fill in wherever it gets thin. If you follow the directions for watering, mowing, and fertilizing it should look great by July. Deep watering is pretty key to success with all grasses.