What is the correct action for worshipers who thought the imam’s takbir was the takbirat al-ihram, then realized it was a takbir of transition (sami‘a Allahu liman hamidah)? And is it necessary to repeat the prayer for someone who performed an additional rak‘ah, thinking they had missed one?

If the worshipper catches the Imam while he is bowing, and the worshipper says the Takbirat al-Ihram (definition) while standing, then bows before the Imam rises, he has caught the rak'ah. If the Imam rises before the worshipper bows, or if the worshipper doubts whether the Imam rose before he bowed, then he has missed the rak'ah. In this case, the takbir for bowing is waived, and the Takbirat al-Ihram suffices for it. If he misses the rak'ah, he must perform it after the Imam says the taslim (definition). However, if he catches the bowing with the Imam, he is not permitted to perform an extra rak'ah after the Imam says the taslim, unless he doubts whether he caught the bowing.