No. It doesn't have to be in sequence, just three indentical positions. But it's seldom evoked OTB out of sequence because most people can't remember exact positions. I read a long time ago where Fischer claimed a draw by rep and it was contested. They had to go through the game score to find the three non-sequential instances. I wish I knew what game that was. Many people online think the 3 fold rep is automatic, and, perhaps, some sites incorrectly apply it automatically, but, as you mentioned, it has to be claimed and isn't required unless it is claimed.
threefold repetition

The game batgirl refers to is Fischer-Petrosian, Candidates Match 1971, game 3. Petrosian contested this, and the arbiter upheld Fischer's claim. Petrosian later admitted that he had missed the non-sequential three-fold repetition.
This rule to me is confusing but not the way I had learned it. I always thought moves repeated three times was threefold repetion but its more complicated:
Fide 9.2 The game is drawn, upon correct cliam by the player having the move, when the same POSITION, for at least the third time (NOT NECESSARILY BY SEQUENTIAL REPETITION OF MOVES).
I'm guessing then that anytime during the game if the same position appears you could claim a draw by threefold repetition. I always thought it had to be in sequence. But I noticed some interesting games about this rule in Wikopedia and wondered if others had comments.