A chat program that allows people to communicate through the internet in real time is known as

Instant messaging lets people exchange typed text over the internet in real time as each line is sent.

Distant messaging is not a recognised term for such live chat services.

A chatting room is a shared space for many users, whereas the described program is the messaging mechanism itself.

Only one term names real-time chatting correctly, so a combined answer does not fit.

Instant messaging typically shows presence, such as whether a contact is online.

Early popular instant messengers included ICQ, AIM and Yahoo Messenger.