I'm not sure I understand the distinction you are making between the terms your children are using and the terms Future's children are using?
The children have two houses "Moms house" and "dads house" They get these terms from everyone.
Dad has two terms "our home" and "Moms house" Mom has two terms "our home" and "Dads house" The teaches, friends,etc have two terms "moms house" and Dads house"
"What is best for my kids is best for me" Amor Fati Link to quotes: https://www.divorcebusting.com/forums/ubbthreads.php?ubb=showflat&Number=2879712