Okay a couple things from my humble opinion
First, let me tell you from pictures of me at the hospital after the birth I looked WAY BETTER immediately after the birth than I did in the next few days. I also felt better because I didn't have time to get sore yet. It really hits you. I am mentioning this because of the diaper thing, you're going to be in one for days, so if people are going to visit early on you just may want to get it over with. Maybe. That being said people will be visiting me the next day unless I have the baby in the morning and feel up to it.
Personally we let people know up front that we didn't want anyone at the hospital waiting because I didn't want the stress, it was just DH and I, and we called them when it was done.
Whatever it is you want, I would talk about it beforehand and make it clear.