Looking for etymology or information on the alternate meaning of "I don't care to X"

I think it's just a corruption of "I don't care about getting dirty", similar to how "I could care less" is often said when the user obviously means "I couldn't care less."

Having grown up in an area where the positive sense of "I don't care to VERB" is common the best equivalent of this in standard English is more at "I don't mind VERBing" rather than "I don't care about VERBing". I have no sense of the etymological history of the phrase, and I suppose it makes sense that it might have come from the 'care about' pattern, but there is a subtle difference in meaning:

I don't care to pet-sit your cat while you are away.

which means the person is positively agreeing to do so even though some people might consider it too onerous a task to undertake. That doesn't have the same meaning as:

*I don't care about pet-sitting your cat while you are away.

I'm having trouble giving a good interpretation but it seems to have either a neutral or mildly negative sense.