Isn’t cheating doing things behind someone’s back?

Husband neglects wife for years (at least 10 yrs), wife going through depression after repeatedly trying to save the marriage and initially asks for a break but eventually talked about each other doing what they want (aka be in a relationship). Wife starts to fix herself after only taking care of husband and kids for 20 yrs. Husband is aware that wife has suitors. And wife was honest about a specific guy giving her more attention. She eventually falls for him, all with the knowledge of the husband. Is this still cheating?