Hi J.,
In our house we have always said "your private". I have a boy and a girl and I got them use to saying your private or private area. My daughter knows my son has a "private area" that is different and all his own and my son knows his sissy has a "private" that is different and all her own.
We teach them to respect that how ever your family chooses to do so. We homeschool our children so it especially compliments us in other words in has not been threatened.
My daughter is 11 and we have been gradually talking about puberty for the last 2 yrs. and just this last week we talked extensively about her referring to it with Doctors and me (Mom) as vagina when needed. She learned that's what it is awhile back but she feels it is not necessary to change it unless she has to. She feels more comfortable still referring to it with as "my private". She actually feels the name "vagina" sounds gross. It kinda does to me too. I heard a 5 year old say my vagina recently and it just sounded "Odd" to me! I mean it was in context but it felt strange to hear such a "Big Word" like that from a little one. That's us. Taken out of context it is a "Strong" word and has been abused way too many times!!!
I will never regret how we chose to handle it and as a parent I think we have made the right choice.
I hope this helps and you find what works for your family.
A. in AZ