Must the prayer be repeated if the worshiper added an extra rakʿah due to an incorrect ijthihad, thinking that the first rakʿah was not counted?

The majority of scholars are in agreement that whoever catches the Imam while bowing has caught the rak'ah. What you did by making up a rak'ah in its place is considered an addition to the prayer. Scholars have differed regarding the ruling on one who adds a rak'ah out of ignorance, and what we deem more probable is that re-offering the prayer is not obligatory. However, it is preferable and more cautious to re-offer it to avoid the difference of opinion.