Praise be to Allah.
If you only uttered the word of divorce (talaaq) on the second or third occasion because you believed that it was obligatory, as you had been told, then this talaaq does not count as such according to the more correct opinion, because if a person bases his talaaq on a reason, then finds out that that reason was not valid, his talaaq does not count as such.
Shaykh Muhammad ibn Ibraaheem (may Allah have mercy on him) said: We have received your letter in which you ask about your divorce (talaaq) of your wife and you said that you heard something about her and you got angry and divorced her three times (threefold talaaq), then after that you found that the news you had heard was false and you became certain that it was not true, and you are asking whether this divorced counts as such or not, because it turned out that she is innocent of what was said about her.
Praise be to Allah.
If the situation is as mentioned and you only divorced her based on this false news, then the correct scholarly opinion is that this divorce does not count as such. Based on that, the divorce is invalid and your wife is permissible to you on the basis of the first marriage contract. There is no need to go through the process of taking her back or do a new marriage contract.
End quote from Fataawa Muhammad ibn Ibraaheem, 11/ question no. 3159
See also question no. 36835
The one who told you that made a serious mistake and spoke about Allah without knowledge.
For a woman who menstruates, her ‘iddah is three menstrual cycles, not three months. If her husband gives her a revocable divorce and does not take her back during the ‘iddah, she becomes completely divorced and is not permissible for him except with a new marriage contract.
As it has become clear that your second and third talaaq do not count as such, but you did not take your wife back during her ‘iddah following the first talaaq, then she has become completely divorced from you and cannot go back to you except with a new marriage contract that fulfils the necessary conditions.
And Allah knows best.