The quick and dirty is this:
Calcitic lime is just that, a calcium-only or calcium-majority lime.
Dolomitic is one that has a good amount of magnesium in it. It can vary, but 1 part magnesium to 2 parts calcium or less is common.
I once sampled Waukesha lime and it was 11% magnesium, and 22% calcium. I got some pell at Menards that was 3% magnesium, 33% calcium. I just picked up some calcitic at Fleet Farm over the weekend that was 1% magnesium, 38% calcium. So you've gotta flip your bag over to know what you're getting.
Calcitic is good on heavy clay soils
Dolomitic is good on sandy soils
In my mind, ag lime is anything that isn't pell lime. There are fineness grades that affect how fast it gets to work, but I never get too bent out of shape about that. I'm gonna use the larger particles in the years to come.