Well, if the depression was years ago, this might not have had any bearing on the affair, but more often than you'd perhaps imagine, people use an affair as a form of self-medication for depression and stress. The reason for the affair lies not within the marriage, but as a response to the chaos and general grind of life. However, this never happens to people who have strong boundaries around fidelity or who are in the habit of putting their partner first. Affairs like this are ultimately borne out of selfishness.
Your posts are quite brief and this might be reflective of your shock, but the affair you are describing sounds more like a feelings addiction relationship. I think it highly unlikely your H is actually in love with the OW, but he might have been infatuated. What is however extremely likely is that he became addicted to the feelings this generated in him about him - and that his feelings for the OW herself, never did run very deep.
If your H had a tendency to depression (and also self-absorption), then the arrival on the scene of someone telling him how wonderful he was etc., was possibly a very intoxicating experience. As time went on, he possibly became addicted to the whole enterprise itself of having an illicit relationship, rather than the woman herself. Hence, his affair was not motivated by sex, but by feeling sexually desirable and valuable, to someone else.
WRT what he is telling you about the physical relationship, I can tell you that in the aftermath of a discovery, it is unusual to hear the whole unvarnished truth, especially from someone who wants to save his marriage. The truth may lie somewhere in between what he is telling you and what you fear the most i.e. there was a physical relationship that involved everything except penetrative sex, because there is often a rather skewed Clintonesque morality in both sexes about what constitutes infidelity. If he went to her flat twice, I would hazard that far more than kissing took place.
Even if he had a full sexual relationship with her, it actually makes no difference to why the affair happened, or your recovery generally. Either way, you are still dealing with infidelity and a massive betrayal of trust.
Lots of affairs happen regardless of the state of the primary relationship and they happen in relationships that were happy and sexual, as you have discovered. It is a mistake to think that this only happens in relationships that are unhappy or sexless, because unless both partners in that happy relationship enforce boundaries in the friendships they make, affairs like this will happen.
It will help you tremendously if you accept that this had nothing to do with you, or even your marriage - and believe what your H is telling you in this regard at least. This affair, like every other, is more about him than you or your relationship.
The challenge for him is to discover what it was about him that led him to choose these actions. Until he discovers that, you won't be able to recover from the affair or have any certainty that this won't happen again.