If eChess doesn't work because all of it is electronic and Correspondence doesn't work because snail mail is not involved then the previous mentioned email-chess should serve as the modern version of correspondence without implying that it's live.
Debate: What to call "Online Chess"...

My vote would be for naming it: TURN BASED CHESS
(I do not know whether is was named in the above 204 earlier postings).
Why: since indeed lots of players would not realy understand the characterization 'Correspondence Chess' + TURN BASED represents is in fact somehow the 'opposite' of 'instant chess/ live chess' which is in use for the one of the other forms of conducting style of playing chess.

I don't know if it's been previously suggested, but "Intermittent Chess" is about the best fit I can come up with.

We still happily use the term 'correspondent' to refer to news reporters who have long since given up the practice of sending their reports via the postal system.
Why is it so hard to allow users of chess.com to take for granted the fact that their 'correspondence' won't require any stamps to be licked?
Despite my little jest above, OCC is a good description. But I guess that the site doesn't want to use an abbreviation and the whole lot is too long, so that just brings us back to CORRESPONDENCE CHESS.
My vote is cast.

"Am/PM Chess"
"Week Long Chess"
Sorry for this one, "Geritol Chess" (spelling is probably wrong)
Do you mean "Genitol Chess"?

...or the more snappy 'corrie chess'. This could also refer to the British drama series which has been running for over 51 years - like some of the tournaments here seem to.

Online Correspondence Chess (OCC)
Obsessive Compulsive Chess (OCC)
lol... perfect.

Correspondence Chess (53%)
- Email Chess (4%)
- Echess (11%)
- Slow Chess (7%)
- Daily Chess (6%)
- Turn Chess (3%)
- Turn-based Chess (17%)
Thank you! 1374 votes cast

my PACED CHESS never made the cut.... damn.
I think cChess should be up there though.. it looks good, sounds good and represents what it really is, and kinda marries in with chess.com.. both being short and sweet.. and just have under it in light grey small lettering (correspondence chess).. to explain what it is, but very subtly.
Online Correspondence Chess (OCC)
Obsessive Compulsive Chess (OCC)