Is it necessary to wait for the muezzin app's adhan to finish before performing the two rak'ahs for greeting the mosque?

What you did is fine, and your prayer is good. The muezzin made a mistake by leaving his phone's adhan on, distracting people from their prayers, and by getting angry when you advised him. Advise the muezzin gently that he should not distract people from their prayers and should turn off his phone's adhan or leave it outside the mosque.