What is the ruling on the prayers of one who was ignorant of the place for the transitional takbirat, and erred in the takbir for standing up from the two rak'ahs, and then learned of this error from someone else, taking into consideration the obligation of the takbirat?

Your prayers are valid, for intentionally omitting the transition takbirat (saying "Allahu Akbar" when moving between prayer postures) due to ignorance does not invalidate the prayer according to the Hanbalis. These takbirat are considered among the wajibat (obligatory acts) that are waived due to ignorance or forgetfulness. However, omitting them necessitates prostration of forgetfulness (sujud (definition) al-sahw), which in turn is waived if a long time has passed.