They're two different games.
Well, maybe you are right. But, no, it appears to me that there is only one game: chess.
Speed, regular play, turn based...these are just different time limits for playing the exact same game. With the exception, of course, in turn based chess the use of a database is allowed.
I can see how one can have a differing rating in these situations. For example, I avoid blitz and bullet because I have found that they breed a propensity for me to play sloppy moves.
With correspondence chess, one has access to opening book databases and analysis boards and a great amount of time. I personally will not play with aid of any kind. My view is that players should learn at least the common open lines and not need an assist. That's just my belief; I can't bring myself to making moves decided from a database.
It is understandable how one's rating can vary form one of these types of gamers to another. But, shouldn't that all be baked into the cake? Shouldn't someone of a certain competence get a rating within a nominal +/- form one type game to another? Isn't a spread of 500...700...900 points excessive?
Someone may play 1000 at blitz, 1200 at long chess...yet be 1800 at turn based. Does this make sense? Seems to me they are really a 1200 player.
Where did I go wrong in my thinking?
"play a kind of game like you've never played before"
So basically,
Want to improve your chess rating? Play better chess!!!
On a serious note though, if you play better players I think you are less likely to pick up bad habits which could bog down improvement.