Is the 'Asr prayer that I performed in congregation after the Maghrib call to prayer considered a qada' (make-up) or ada' (on-time) prayer, knowing that the congregation had already begun their prayer before the 'Asr time had expired?

The adhan (definition) heard after sunset signals the departure of Asr prayer's time, making the prayer a qada' (performed after its prescribed time). Performing a prayer, either in whole or in part, within its prescribed time is considered ada' (on-time performance), whereas performing it after its time has elapsed is qada'. Your Asr prayer in this instance is qada' because it occurred outside its designated time. However, your joining a congregation performing Asr as ada' is permissible according to some scholars, such as the Shafi'is, and you are not required to repeat the prayer.