If a rightsholder could sue you for the first case, they would. The difference here is you're transmitting it:

• Over the internet

• In the form of encoded audio data

And there's legal precedent here that says this is copyright infringement.

And yes, recording the song would be trivial, though it may not be of good quality.

Granted, they probably don't care that you and a small circle are listening to a song. Music publishers only go after big fish like distribution networks, ISPs, and repeat infringers, only taking the time to own small individuals when it's an open-and-shut case for the sake of making an example. It's not worth the money to sit and spy on every small Discord server to make sure nobody micspams.