To be honest, those don't sound like chicken pox - they usually don't disappear in a day. They start off like a red bump (like a forming pimp), form a clear center, then crust over. The spots themselves last for several days to a week or so.
As far as the shot. All of my kids have been vaccinated. My eldest was vaccinated near around when the shot was originally released to the public. She came down with a moderate case of chicken pox. My feeling was hopefully it prevented her from having a severe case. She has been exposed several times since then, and hasn't had them again, so she's not one of those kids who never gets immunity *thank goodness*! She was 1 when she had the vaccination - around 3 when she had chicken pox.
My younger daughter looked like she might have one pock mark *maybe* around the same time that her sister had chicken pox. That was a few months after she'd been vaccinated (twice by accident).
My son was vaccinated at 12 months. Challenged since, and no problems.
I know of *little* babies who have had chicken pox as well as adults who get it. There's really no age range. The idea behind the vaccination isn't necessarily 100% immunity, but to prevent the complications associated with chicken pox and hopefully to prevent shingles down the road.