Is my prayer valid, knowing that I took my phone out of my pocket to turn it off during the first rak'ah of the 'Isha prayer, and then bowed after the Imam had already risen from bowing in order to catch up with him?

If the Imam bows and rises from bowing while you are still standing because of your negligence in turning off your phone, there is a scholarly disagreement regarding the invalidity of your prayer. The Hanbalis consider the prayer invalid if the ma'mum (follower) lags behind the Imam in bowing without a valid excuse, while many jurists, such as the Shafi'is, believe that the prayer is not invalidated in this case. Therefore, if you perform the prayer again as a precaution, it would be better, and there is no blame upon you if you act according to the opinion of those who do not consider re-performing it obligatory. However, if you had a valid excuse, your prayer is valid by scholarly consensus.