I do think of it as cheating. If you google "emotional affair," there's a Wikipedia page for it.
I've been the cheatee in an emotional affair during a long term, monogamous relationship. My ex was depressed. He became deeply, emotionally involved with a colleague - 3 am emails and chats, 2 hour long phone calls, involvement with her friends, family, and kids. He withdrew physically and emotionally from me. He didn't even want to tell me about his day. This went on for months. He claims they never had sex. Frankly, I'll never know, but it didn't really matter. It ended our relationship.
One take on my situation is that we had different definitions of cheating. However, his inability to love us both at the same time, monogamy or not, was a real problem.
I've read that women are more hurt by their partner being in love with someone else and men are more hurt when their partner has sex with someone else. That is perhaps why you and she see the situation differently.
That said, plenty of couples deal with sexual cheating and manage to repair their relationships. Some may even make their connections stronger with the honest work they do to get there. There's no reason why you and your wife can't work this out, too.