Babička chce šálek kávy. Why is not correct The grandmother wants a cup of coffee? Why is correct My grandmother wants ... ? .
There is no such meaningful distinction, both grandma and grandmother are babička.. babča is a very colloquial, or even insulting, word (I never use it) and it is not found in normal dictionaries (PSJČ, SSJČ, SSČ).
I stress again that "grandmother" IS accepted here.