Is this instance considered the third divorce, thereby rendering the husband forbidden to his wife, taking into account the previously mentioned divorces?

If a husband divorces his wife three times in one go, it is considered a revocable divorce according to Shaykh al-Islam Ibn Taymiyyah and those who agreed with him, and it counts as a single divorce. If he takes her back after this divorce, he still has one divorce remaining, after which the major separation (baynunah kubra) occurs. If he divorces her a third time after that, either verbally or in writing with the intention of divorce, she becomes unlawful for him.