Most Helpful Guy
Obviously, he is effort to text and communicate with her behind your back is a strategic effort to avoid the appearance of impropriety. Thus, its a clear sign that he is fully aware of your feelings and boundaries on the matter and the impact it could have on your decision to be with him.
He has feelings for her, end of story, and is simply unready to walk away from his past and current relationship/friendship with her. Yet, he does not want to end what he has begun you. He seks the best of both worlds.
So, what can he do? What does he do? He takes upon himself to never speak of her name to you or communicate with her in your presence to mislead you. As such, it's deception.
Now, you should decide whether you can forgive, trust and can be with someone, who, unfortunately, is yet to completely get over his ex. For regardless of whether you leave or remain, she will be on his mind until he is over her.
Most Helpful Girl
First things first you said this is my second series relationship... But I thought this wasn't a serious relationship to begin with. You may think that's real minor and you may very much be in love but that sets the tone of the relationship right off the bat. Obviously he still cares for the girl as he should, no offense. But if thats a boundary you DONT want him to cross. Tell him that's off limits. If he left his phone in the car with you then he trusts you and shouldn't really care if you looked though his phone. I say bring the situation up.