No, puzzle rating doesn't have any correlation to rapid or blitz ratings. It just measure how good are you tactically. It is pretty normal for puzzle rating to be a lot higher than other ratings.
Chess ratings across different games
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On this site I am rated around 1250 playing 15/10 chess, 950 in Puzzle Battle, and 1600 in Puzzles. This seems like a really wide variation in ratings. Is it normal (which means it's easier to get a high score in Puzzles because there is no opponent), or does it suggest that time pressure is especially destructive to my chess/tactical thinking? Obviously time pressure is bad for everyone, but the 950 in Puzzle Battle, and the 1600 in Puzzles suggests it's worse for me. If I do especially suffer from time-induced poor decision making, is there any particular training to improve this weakness? Thank you