Is the new contract considered a new marriage or a return [to the previous marriage], after the man divorced his wife and did not take her back during her waiting period, and then contracted with her a year later?

If a man divorces his wife with a single revocable divorce, and he does not take her back until her iddah (waiting period) expires, then he marries her again with a new contract, this is considered a new contract, and he still has two divorces remaining. She becomes unlawful for him if he divorces her twice after this contract, because he would then have only one divorce left, after which she becomes unlawful for him with a major irreversible divorce (baynunah kubra).

If he divorces her with a revocable divorce, he has the right to take her back before her iddah is complete.

A woman divorced with an irreversible divorce (bain) before consummation cannot be taken back (without a new contract). If he wishes to marry her again, it must be with a new contract, and she returns to him with two divorces remaining. If he divorced her twice and then married her, she returns to him with one divorce remaining.