I think the problem defining the term comes from the fact that being a parent at home comes in so many different forms, that most terms usually attempt to describe an aspect of it, so that it may not apply to everyone.
A) Some women stay at home and spend 100% of the time looking after their house and children.
B) Some women spend their time looking after their children, but have cleaners so do no housework.
C) Some women are at home, but their children are at school, so do not really do much childcare (Esp if their children are of secondary age).
D) Some women are at home, but do very little childcare or housework because they have au pairs, cleaners and nannies to help them.
E) Women who are at home, and do some childcare, but not all the time because their children go to preschool, nursery or the crèche on a regular basis. These women may or may not also have cleaners.
F) Women who stay at home during the week, do all the childcare and housework and then work one or two evenings a week in a little part time job when they're children are asleep (friends who do this do consider themselves SAHM) even though they do work. There are also those who have a little job working at home, but keep their children with them.
I am sure there are women who do not fit into any of these categories, but these categories describe some of the very different experiences of women I know who stay at home.
How do you find a word that describes it all? Homemaker doesn't really apply to B, D and sometimes E. Full time parent doesn't apply to C,D, E and sometimes F. SAHM doesn't apply to any of them if the women are out a lot.
The term that seems to cover it all, seems to be 'based at home', given that the main objection to SAHM is the fact that many women are out and about a lot. All women who do not work are based at home - that is their default place, when they're not out at their various activities and it is where they return to during the day. But, this doesn't really work because a) it sounds contrived and b) it doesn't actually say anything about what the woman actually does (which I think is some of the objection to the term SAHM).
However, given all the very different experiences of women who do not work, I do wonder whether it is possible to get a term that covers all those variations and is actually meaningful. Somehow, I doubt it.