Moderating online education forums: delete, snip or leave rudeness?

Snip the last part out, send a private message to the poster reminding them of the behavioral standards expected in the course forum.

I don't see any reason to remove the whole post, if there are parts that are appropriate. Removing the whole post makes it seem like your motivation is to punish the poster, rather than to keep the forum professional and constructive.

I would recommend against leaving rude comments there until/unless someone takes offense. This is a forum that is hosted and moderated by you, not your students. It is your responsibility as an educator to create a constructive learning environment (to the extent that you can). It's not your students' responsibility - they have to be responsible for their own actions, but they shouldn't have to be responsible for making you respond to other students' actions.

You can also consider imposing a stricter rule where you remove all comments that are not directly related to the subject matter. You can also let the first comment that goes off topic stand and then intervene in such a thread by saying "the previous comment strayed off topic, let's get back to discussing the topic". That comment that strayed off topic can then serve as an example of a conversation that will be stopped on the forum.

An email list that I'm a member of where researchers discuss a specialized topic is moderated in this way. Sometimes someone gets carried away and starts to discuss politics that's tangentially related to the topic matter. The moderator of the list will then call on the participants to stop discussing this, but this typically happens after a few such posts.

By keeping a short leash on anything that deviates from the topic, you make it difficult for rude comments to creep in. The rule is easy to apply uniformly, if you only intervene when there is a rude comment, then you may be accused of unfair moderation as there may have been similar comments in the past that you've let stand.