re: How much does it cost to pour 200-300 sq ft of concrete for a patio?

As most have said the small amount of concrete drives the price up. You are taking less than a half loaded truck which runs up delivery costs. You indicated that the area is dug out but is it formed? Typically you don't just lay concrete against earthen sides, earthen bottom OK but not sides. Are you doing the finish work or the contractor. Are they including welded wire fabric? Would highly suggest for just 4" (6" by 6"-16 ga.). How far are they going to wheel barrel the concrete to your back yard. Your looking at 20-30 wheel barrel loads. Some mentioned a pumper truck which IMHO does't work for this. First a pumper typically cost $1000 plus and I don't think that 3-4 yards of concrete is enough to fill the tube or to prime the pump and have it working properly.

If you think this is too much to pay then I suggest that you look to a local rental place. You can rent a small concrete mixer add quick create, water and rock right next to the patio and place it yourself. If you go this route I suggest that you buy the WWF 6x6 and do your best to make sure it is in the middle of the concrete. Sitting at the bottom does you no good. Also, when finishing the concrete I would suggest a maximum of 8' x 8' panels which can be achieved by a soft "cut" grove in the concrete when finishing. Just remember that concrete will crack and you want to control where that happens.

If TL;DR....It's not a bad price for a small quantity. If you can do it cheaper yourself go ahead.

No. You aren't going to find a concrete finisher that has a license. Not for 300 feet.