Is my prayer valid if I entered the mosque and found someone praying, so I completed my prayer behind him, while there was another congregation praying in a corner of the mosque, and I refused to leave my prayer to join them?

What you did, praying behind someone who joined the prayer late (masbuq), is valid and there is no issue with it. It is not prescribed for either of you to break your prayer to join another congregation. This issue is a matter of scholarly disagreement, so there is no disapproval for whoever believes that it is not valid to follow a masbuq. If the brother who called you to break your prayer intended for you to join another congregation, then it is not proper to establish more than one congregation in the same mosque at the same time, but the prayer is not invalidated if that occurs.