I think we need to look into the exact purpose Rule 2 serves to really answer this question.
2. Do not use information gained outside of in-character means.
I.e. metagaming. This especially refers to communication between players outside of the game via things like Discord, known as metacomms. Characters are otherwise allowed to know everything about ingame mechanics or antagonists, as well as keep persistent friendships or relationships with other characters when not for the purpose of unfair advantage by teaming up together for little IC reason.
Rule 2 serves explicitly to stop a player from using outside information to gain an advantage in-round. If a player is not playing the round, or straight up does not play space station 13, then I fail to see how they could use that information in the first place.
With that said, to answer some of the scenarios above with my own:
1. Streaming to your Girlfriend that has never played Space Station 13 in DMs is obviously OK. She's not going to use any information in the round, because she would have to be playing on the servers in the first place to use the information.
2. Streaming to a friend group's server that has literally never heard of Space Station 13 and you bring it up as something interesting to show off. This is also fine, because again, nobody in that call is playing Space Station 13- or on TG for that matter either. They can't use the information.
3. Streaming in a server with other TG players... This is bad. Because you really have no guarantee of who is going to be watching your stream. But again, as long as the people in the call aren't playing the round, then is there really any impact at all?
Now here's the thing, all of these scenarios are OK by the literal definition of the rules. No-one is actually
using outside information gained outside of in-character means. But the issue really becomes about responsibility. What happens when one of those friends in the call start playing the round? If you ask everyone if they're in the round, and they respond with "no" but actually are, whose responsibility does it ultimately become? Where's the line to draw for irresponsibly streaming to people who could very well use the information? IMO you definitely have a responsibility as the streamer to make sure no-one in the call is currently playing the round. It's a case-by-case common sense thing (gasp) that admins gotta make the call of when handing out bans for people being Stupid with it.