ice water and ocicats

On the train today I was pondering how the English language could really use two different words for two usages of the word "my". There's the possessive my (my purse, my bagel), and then there's the "associated with me" my (my company, my train, my husband) where I definitely don't own or control the thing in question. There are more, but these seem like the two that are most in need of separation.

I have no idea how my hypothetical language change would handle things that one partly owns or partly controls.

My cat: I own her under the law, but she also has legal rights to humane treatment, and I think of her as more than a possession.
My condo: I own half of it at most, and that's joint ownership with heisenbug involved, the bank owns the rest.

Is this even interesting? It seems kind of unactionable. There are ways to protest the use of a word, or the misuse, but there's not much people can practically do to promote a pair of new words to replace one word. :P Also, I love wordplay... I am exactly the kind of person who's interested enough in ideas about clarity of language (computer languages, Esperanto, even E-prime) but then goes home to do cryptic crosswords.

Maybe I'll do some constrained writing soon.