The way the Recently timed out marker works is this:
It does not apply to live games. Only correspondence games.
That message appears on your profile after you time out of a correspondence game. It also means, that all subsequent losses by timeout on correspondence games will be silently annulled.
That message only goes away if you end a correspondence game by scoring or resignation.
Your last correspondence game was https://online-go.com/game/11931173, which was in 2018, which ended in you losing by timeout.
Since you only play live games, if you are really concerned about that message, to get it to go away, you can play a correspondence game against a bot. They will move quite quickly. It doesn’t even have to be ranked, as long as it ends by scoring or resignation.
Hope this helps!
Edit: darn, @Conrad_Melville posted like a minute before me.
I don’t believe it has to be a ranked game to remove the marker. I wouldn’t be surprised if it was, however, because then you could easily game the system by playing a bot unranked, then resigning after like 3 moves.
If it does have to be a ranked game, I believe you would actually have to play it out, as annulled games would be just as much “cheating” as unranked games.
However, unranked timeout losses and annulled timeout losses can definitely trigger the thing.
I don’t see why everyone dislikes this message so much. The way I see it, the correspondence recently timed out mark grants you immunity, so in case you do decide to play a correspondence game (which it looks like you won’t), and in case you do time out, it won’t affect your rank.

