@Nothappy, I looked at your threads and saw that you had occasion to look at his phone around twice in a year.
Is this the situation? Initially he did not mention this woman’s presence at a work meal, but did later, and you felt a niggle. You investigated, and she was not on his message list as others were, and you asked about this.
He said that he deleted her texts (and only hers) because she is attractive, they get along well, and “he thought it was better to delete them than risk upsetting me...”. I believe he said that you wouldn’t care for their banter. He apologized, and admitted that he would not like you to delete texts from males. You asked him to stop deleting her messages. You admit feeling jealous and anxious at times, but you do feel that his actions in this case were inappropriate.
An emotional affair would occur if he secretly channels emotional energy, time and attention into someone other than you. He did keep their communication a secret from you. You never saw the messages, so you don’t know their content or frequency. However, something was different about her messages. He didn’t delete those of the other women.
Is he now describing their relationship as “close?” If so, then it could very well be an EA. Personally, I would wonder what else he is hiding.