I just have a couple of thoughts.
First off, I'm glad your O.K., and this really shows the importance of a ROPS in the up postion and your seat belt on.
I also mow a similar hill, and it may be steeper, and I have also forgot to put mine in 4 wheel drive and slide down the hill. I don't have anything at the bottom and have a little flat area, so its a little safer.
You actually can regain control by speeding up. The rear tires will "catch up" to the ground speed of the tractor and give you more control, but this isn't the instictive thing to do. I used to mow my hill with a 2 wheel drive riding mower, and this is how I went down the hill, I'd start out slow and gradually speed up to keep the rear tires moving.
I don't think the kind of tires had a lot to do with it. The grass looks pretty lush and just cutting green lush grass can makes things a little slick.
Someone else mentioned think about what you will do if you bump your tractor into nuetral. I had that happen to me on my hill and it wasn't pretty. If you are used to driving a HST tractor you don't use the brakes a lot. If you bump it into neutral, the brakes still work but I had trouble getting on the brakes quick enough. The tractor free wheels and picks up speed in a hurry on a hill in nuetral.