Holy cow look at that sand. Visual confirmation of your soil test. Bermuda loves sand.

From the pictures it looks like you have two sources of water. One from the down spout in the first picture, the second, in the center of the first picture, I can not see a source. From your pictures it is difficult to see the total problem. It appears the ground, a half dozen feet from the patio, is higher than the ground at the downspout.

Regardless of what grass you plant the first thing you need to do is channel the water and control its discharge. What you can do depends on the size of the area being drained.

Looking at all of your lot you need to determine the natural contours and water flow. Then try to re establish those natural flows. Before the house was built did the natural flow go through the area where the house is now located? Did it go towards the front under the fence?

If it were me I would get a lawn chair and sit on the patio and spend and hour looking at the area. Not looking at it thinking how bad it is, but where the highs and low spots are in the yard. The elevations of your yard in relation to the highs and lows in your neighbors yard. Is there a storm water drain just beyond the fence? If there is little slope in the yard you may need to get a 10' - 2 X 4 and a level and "survey" the lot.

For the down spout one solution would t bury corrugated plastic pipe and direct it to a safe place near the edge of you lot, but this should be done after you understand the the natural flow. If it is going in the proper direction, the trench for the pipe is already dug.

Yes i plan to seed bermuda in the backyard. Also i have sat and watched the water when it rains. The downspout is one area and also the water comes from under the fence from the front yards downspout located about 3 feet away from the fence. I figured if i hooked those up to rain barrels it would help with the washouts. There is no storm water drain behind the house but there is a area that collects the water out about 20 feet from the house that goes god knows where. I thought about using the pipe but the neighbor next door had his put in and that thing clogs up like crazy and water has a hard time passing through.

Can you run the pipe to the front of the house, and collect the front and back down spouts in that pipe and direct the water to the front of house?

As far as those pipes clogging there is one solution. Take open the pipe at the house and run water through it to flush it out. You may have to run something in it to break the clog (run a hose up the pipe and with a spray nozzle? ). If there is as much water as it appears you will not have to worry about it as the water will keep it clean. At least that what I have found. I have two pipes for two downspouts. After every heavy rain there is always some garbage that is flushed out.

To cause that much erosion, either there is a lot of water coming across the yard at one time OR the erosion is due to years of neglect. If the erosion is coming from years of neglect, that can be corrected with a rake and a shovel. It represents the amount of water you are going to have to redirect it somewhere, if grass is ever going to have a chance.