Are you immune to chicken pox and smallpox if you get cold sores?

When my mother was little, she used to get cold sores (fever blisters, , HSV-1 what ever you call them) all the time, and her pediatrician told my grandmother that she may be immune to chicken pox and small pox. She still gets them, of course, as do I. Just on my mouth and sometimes inside my nose, and they are defidently hereditary. Does this mean my mother and I are immune to chicken pox and/or small pox? I never got the vaccination for either one, but when I was in preschool, my entire cl came down with chicken pox, and school was closed, but I stayed healthy. Was that just pure luck? Or am I truly immune? Sorry if this is a really stupid question, thanks for all your answers!