We have told him that other people will think less of him if he swears, that some words are attacks on groups and hearing these words will cause emotional discomfort to members of these groups, and that his swearing causes his mom some discomfort. I have told him that while I am not inherently bothered by him swearing, I don't want him to do it around me because it will make it more likely that he will swear at school, but he claims that his swearing at home doesn't increase the likelihood of him swearing at school.

A few thoughts:

It sounds like he's being rebellious. Separate the rebelliousness from the question of profanity, and discuss them separately. You might say something like "Asking questions if you genuinely want to understand something better is great, but asking questions to try to frustrate or annoy me is not. I'm getting the sense that you're doing the latter."

If he persists, put your foot down -- but be really clear that it's for the intent to annoy you, rather than because he's asking questions in an attempt to honestly understand something.

It may also help to ask him, openly and gently, why he's being rebellious. Sometimes rebelliousness comes from a perception that the rules are arbitrary or unfair. If you understand what he's feeling, you can be in a better position to address those underlying causes. For example -- Larks's suggestion that sharing the reasons you don't want him to swear may help. And maybe it would also help him to explain how this is a subjective issue, highly dependent on things like tone and social context, and perfectly clear rules are unfortunately impossible.