Is the waiting period (iddah) considered to have begun the moment the husband said, "My role as a husband is over and our marriage is over," and informed his relatives that he had divorced, or is it necessary for him to explicitly say, "You are divorced," for the divorce to be recognized?

The waiting period (iddah (definition)) of a divorced woman begins from the moment the divorce occurs. Divorce takes place when the husband pronounces it explicitly or implicitly with the intention [of divorce]. The husband's statement: "My role as a husband is over, and our marriage is over" is an implicit statement that could signify divorce or something else. If he intended divorce by it, then it occurred.

So, let the wife ask the husband if he intended divorce by that phrase. If he admits to it, then your iddah is counted from the time that statement was uttered. If he denies intending divorce and says he was lying when he informed you of your divorce, then no divorce has occurred, and the iddah does not begin except with an explicit divorce.