MarkoH01: Are you sure that this also happens without using the client? I thought that if you don't use Galaxy you will be displayed as being offline when playing a game. Of course it' would still not be a good thing that you are not able to turn it off but so far I was not aware that the gaming status will be displayed no matter if you have Galaxy running or not. Probably might test this later.
Looks like my wording was confusing, so let me rephrase that with an example.
Let's say you use the client, and I don't. If I go into my chat on the site, and you have the client running, then you show as online to me, even if you're not logged in on the site. Now, if you start playing a game on the client, your status in my chat changes and states which game you're playing.
At least that's how it was from the introduction of the NAS and until the chat revamp, and regardless of if we're befriended or not; it may well still work the same (since the revamped chat was first designed and implemented for and in the client where apparently this sort of thing is regarded as a feature), I just haven't checked as I'm not in the habit of stalking people.