I believe your concern should be the number of plies (belts) instead of tire type, however I do think R4 are far better then turf tires for this application. Example, if I remember correctly Kioti R4 tires are 6 plies were Bobcat tractor R4 tires are 8 plies. In general the more plies the more punctures resistance the tire is. The ply type also affects puncture resistance. I really do not believe AG will buy you anymore puncture resistance then R4, it always seem to me punctures occur between the tread or in the sidewalls.
I did get one small wooden stick in my rear loaded tires. The stick didn't go through the plies but force itself between the top rubber (tread) layer and the top of the plies. This created a small bubble in the top of the tire between the tread. I used a pair of needle nose pliers to pull the stick out. I cleaned the hole then pushed in a plug (glue repair strip) to prevent dirt from entering the hole and grinding on the plies.