True there's ambiguity there and the sentence alone really only conveys that the topic isn't one that the stater wants to focus on any further at the moment.
Generally however, I think there's some assumption or perhaps prior demonstration that implies a lack of care to start.
It's tough to describe. "I could care less what you do with your money" may come with a long understood history that not much care was given to this topic the past (i.e. we haven't discussed it before), and if we are now arguing about it because because some prior discussion escalated, I am indicating that I preferred the status quo with that statement (albeit in a way that probably doesn't defuse tensions in this particular example :).
"I could care less" can mean "I could (stand to) care less".
As in, "The amount I could care about this is zero because it's so irrelevant, but in responding to this discussion or otherwise engaging the topic, I have exhibited a bit of care, albeit undue."
For fun, I sometimes like to propose this to anyone who brings it up, but I could care less whether people say "could" or "couldn't". :)
Generally however, I think there's some assumption or perhaps prior demonstration that implies a lack of care to start.
It's tough to describe. "I could care less what you do with your money" may come with a long understood history that not much care was given to this topic the past (i.e. we haven't discussed it before), and if we are now arguing about it because because some prior discussion escalated, I am indicating that I preferred the status quo with that statement (albeit in a way that probably doesn't defuse tensions in this particular example :).