I think we can all agree that's it safer but there's nothing concrete to say that it's safe. Maybe that's where your families skepticism is coming from?
Regardless, you've taken steps for the best and have done something very very difficult so they should be more supportive.
Do you have to vape in front of them? I stick to one room in the house and pretty much leave the windows open all the time. If I'm out, I'll sneak around the corrne somewhere. I was a conscious smoker and always mindful of other people so I treat vaping the same way.
I had one moron point out that vaping was bad for you as she stood outside the hospital in her gown smoking. I pointed at her cigarette and said "Cancer" then pointed at my Ecig and said "No cancer". She soon shut her pie hole.
Don't get discouraged. Vaping even in the last few years appears to be getting more and more acceptable. Just stick at it