For example: Can I quote a PM in a public post?
I think the instant message is basically the same with private message, but the later is broader. Do you think we should rename it?
I read an interesting statement on the internet today:
Companies are changing their terminology from 'private messaging' to 'instant messaging' or 'direct messaging' because we all know those messages weren't really private.
It was probably just a joke/shower-thought, but I think renaming instant messaging to private messaging won't help much. People will be confused because there are basically three words used across the internet, all to describe a similar sort of online-interaction.
And lo-and-behold: IPS already has the online-interaction tag, which states:
Questions related to interacting with others online, including via chatroom, Facebook, Skype or any other method of communicating online. Can also be used for long-distance-relationships, or just friends you have met online.
If anything, I think we can safely get rid of instant-messaging and only use online-interaction for now. If you're using a specific form of online interaction that e.g. has a character limit, it's a detail that needs to be in the question, not the tags. Looking at the questions now, some are even tagged both online-interaction and instant-messaging, which seems like a double serving of tags to me.
I disagree, I think they have different implications which would benefit from having distinct tags.
As pointed out by the answers to your question, the fact that it was a private message is very important to the situation. Instant messaging is a different thing, and while it does often carry some expectation of privacy, it doesn't have the same weight as when a person specifically chooses to message you out of sight of the "public".