There's never been a set name for it. You may know it as forum rp, play by post, or chat rp. The problem with all of these is that they overlap with other forms of rp. No one is sure if you're talking about MMO rp or ttrpgs.
The essence of it is there are few to no rules for it. You make a character and write about them with other people's characters in a back and forth fashion.
It is due to this form's core nature that I suggest we refer to it as the following.
Outback Steakhouse RP
No rules, just write.