Did you have an IV for your other birth(s)? If so, and you remember getting IV fluid after your baby was born, it had the Pitocin mixed in it, for the reasons your doctor stated. Otherwise, it's given as an intramuscular shot, like a vaccine. It does cause cramping for some moms because it's causing the uterus to contract---just like it does during labor, but this time to prevent bleeding, not to augment labor. Not sure if you can remember back, and if you didn't have an IV or a shot, you must not have gotten it, but in hospital births it's generally standard.
As a postpartum nurse I've had a lot of moms not realize that the bags of IV fluid they're getting immediately after birth have Pitocin in them and had to explain why it was given. That being said, when I was new to postpartum I didn't understand why ALL moms got it, and I tried looking for any evidence in the literature that would lead hospitals to make it routine, and I have found none. It's simply a precaution, regardless of how at risk for hemorrhage a mom is. But you have the right to refuse it, or any other intervention you're not comfortable with, during your birth experience.
my midwives had pit available at my birth center, but I did not need it, so none was used. As far as I know, it is fast acting, so can be be administered if you show signs of need in the moment. as the other person said, you have the right to refuse any and all medical procedures unless you are incapacitated. I would say upon checking in that you expect to be told of any and all medications that are put into iv fluid before administration at any time unless it is an emergency. I would get it in writing from the head nurse that this would be respected by any incoming shifts of nursing staff. The nurses are the ones to administer, the dr. is the one who orders it, but some nurses may just do it routinely after the birth for all moms.
Ask your doctor about techniques to stimulate the natural oxytocin that you produce "to assist the uterus in clotting the placental attachment point postpartum." (wikipedia). I think I read that immediate skin to skin contact with the baby placed directly on your chest and the baby "crawling to the breast" (look up on YouTube) would naturally encourage in rise in your oxytocin levels and therefore aids natural clotting. (I have no medical training, I've just read a lot.)
Ask your doctor what would the medical impact be if your delayed the IV of Pitocin by half an hour? This delay approach sometimes helps negotiate with the medical profession who don't often respond to direct confrontation. If you still need the Pit after a delay then at least you would know it was for medical reasons not procedural ones.