Even if my actions may show it, he says it's regret.
What he might be seeing from you is shame and guilt. That is not the same as remorse, but I can understand how that might be confusing. You feeling horrible for what you have done, feeling sad, angry at yourself, maybe depressed all may seem like remorse but in reality haven't quite hit the nail on the head. You see, all of those feelings are still very self centered. I don't mean that as in selfish, I mean that as in those feelings are self based.
For example, when he expresses his pain and anger do you have a tendency to look down, become quiet, sad, maybe cry? Does your body close in on itself? If so, this is more shame than remorse. In the beginning, it's very difficult to understand the difference. It's my belief that remorse comes in bits and pieces. It's not something that you can have all at once. Which honestly, is a blessing because true remorse can be quite painful.
As you begin to comprehend the magnitude of what your affair has done, you can then begin to demonstrate true remorse. As you examine aspects of your affair, you can begin to "get it". When you get it, you can own it. Getting and owning it is the key to remorse. You can't feel something until you really feel it. Until you can truly attempt to place yourself in his shoes and understand (to the best of your ability) how he feels. An affair in general is too broad a thing to feel all at once. But you can demonstrate remorse for aspects of it.
For example, you may have talked to your AP on your BSO's birthday. Now his birthday may feel ruined to him. Apologizing specifically for this action will demonstrate an understanding for damaging this aspect of his life. Talking about it, explain to him how you understand that it was thoughtless to disregard him on his special day and that you understand that must make him feel second in your eyes. That you want to do what you can to make up for that in the future because he deserves to have a day that is special and just about him without having to think about how much you hurt him.
When "aha" moments come be sure to share them with him. Share your feelings, don't attempt to stuff them and pretend they aren't there for the sake of a good day. Initiate conversations about how the affair has affected the both of you and what your plans are to change to make him feel safe. Reach out to him when he is pain or distant. Don't wait for him to let you know that it's ok to get close.
There will never be a day that you can truly understand how he feels. But with remorse you can view the actions of your affair and feel it through his lenses (perspective).
Don't be afraid to feel his pain. Just be careful not to let it paralyze you. There is a long road to travel still.