For me, as well, partner is the term of choice. I have one lesbian friend who uses the term friend, but only in one situation: the few places where she is not out, like with her 80-something mom. We live in a state where our rights are protected and we are out in our lives. I suspect that matters to one's decisions. I do not mind being seen as lesbian. In fact, I now choose bisexual as my sexual orientation since I am sexually attracted to my partner whom I consider to be female (as well as trans).
We have 4 kids, the youngest is 19. The three older kids are from my first marriage and have always used my partner's name, so changing the name was no big deal to them, though they enjoy sometimes calling her Mama Chelsea, or addressing letters: Dear Mommas. I do not have a problem sharing the title of Mom. It was different when my kids step-mom wanted to be called Mom. I asked that she discontinue that (and she did). In that case, I felt that the biological connection needed to be honored. The kids were very young at the time, and I wanted them to know who the bio-mom was. But now, sharing the title of Mom with my partner who was their biological dad is a non-issue for me. She is the female parent and our language defines female parent as Mom. I do not need to defend it. I am secure that my children respect the fact that I carried them in my womb and she did not. But I will add that she did change diapers, make tons of peanut butter sandwiches and was the one home when the kids got there after school. I had a bookstore to run; the kids came there after school too, but I was usually too busy to give them much attention. If Chelsea had not been a stay-at-home dad, I might be more unwilling to share the Mom title.
Our youngest, whose biological father was Chelsea, likes to tell new kids he meets that he has two moms. Watching the reaction of the people gives him information about them. Some just ignore the comment; some say, wow, that's cool. What's that like? He befriends these curious ones with more enthusiasm, though he is a big hearted kid and tries to help others who struggle with the concept as well. I do agree with those who have said that the children need to find the terms that they feel comfortable with. Chelsea shared with our son that it made her feel good to be called Mom, Momma, Mamacita etc but just once. He took it from there.
Peace (a farewell salutation I picked up from our son...),
Ruby