Great suggestions and points made my PPs. My DH is a gamer too, and we've had similar situations.
About 10 years ago, I was flipping through TV channels and came across some guy on a stage lecturing an audience. I think he was trying to sell a book. I don't remember who he was and I only watched a little while but he told a little anecdote that was very eye-opening to me and I've often thought of it in situations like this.
His story was something like the following. Sally wants to spend some time with her husband and a new movie is out. She says "DH, let's go to NewMovie together tonight." Her husband thinks about it but he doesn't really want to see NewMovie. He wants to see OtherMovie. He says "I don't want to see NewMovie. Let's see OtherMovie." But Sally's not interested in seeing OtherMovie and she says so. So her DH says to her "OK, well, Bob would probably like to see OtherMovie" and he calls Bob and they go see OtherMovie. Sally is left alone wondering where she went wrong.
The man giving the presentation had a list of 5 "layers" of desire or some such thing. The only two layers I remember were the two mentioned here. Sally wanted to do something with her DH. She just wanted to spend time with him. It was the relationship layer. It didn't have to be that movie specifically, but she didn't communicate that to him. Her DH heard her suggestion but only heard about the activity. That was lower on the priority list than the relationship layer; it was the activity layer. Her suggestion planted in his mind the desire to see a movie. And the specific movie, to him, was more important than who he shared the experience with.
Sometimes, Dark Magdalena, I'm in your shoes with my DH. And remembering the thing I saw on TV, if he asks me "what do you want to do now?" I will think about it answer something like:
1) Activity priority: "I want to go on eHell and catch up on some discussions. If you'd like to sit beside me and watch TV while I do that, it would be fantastic. But if you would be bored, please, do whatever you're interested in doing (here or in another room). I'm going to surf eHell."
2) relationship priority: "I want to spend time with you. Ideally, I would like to surf eHell, but if that bores you" (or bothers him because my chosen activity has sound or too much light or something) "then what would you like to do and I'll do that with you? For example, if you want to watch the football game, I can find a book and sit beside you while you do that."