https://www.ecfr.gov/current/title-14/chapter-V/part-1221
So you wouldn’t exactly get a copyright claim when abusing the NASA logo but it’s still illegal.
I couldn’t find anything about the NASA word itself though, just some articles reciting guidelines by NASA not to imply an endorsement by NASA. I don’t know how that’s enforced though.
Those "independent streamers" provide live launch streams with multiple feeds using their own equipment and to top it off they have numerous very knowledgeable hosts for all their streams. At this point I suspect they are covering every US based launch from all the major players. Hell, today they broadcasted both the New Glenn and Starship launches less than 24h apart.
But yeah, let's get hung up on an organization name that originated as an Internet forum for discussing all things....... NASA!
You are allowed to basically reproduce the work without any worries whatsoever.
Sure, you'll get better telemetry info and the onboard views from the ships that these companies launch in their streams, but the commentary is sub-par at best (they are always sounding so "corporate official" to me) and they just don't provide the best views for watching it live.
I love that these space flight companies have opened up their development process to let the public follow along, I just think they aren't as good at producing live streams as some of these channels that have taken off over the last 5+ years.
They started doing it when SpaceX was launching their first fuel tanks literally in the middle of nowhere, you can just sit on a side of the road a few hundred feet away and record (or even stream) everything from a basic Webcam. Eventually more and more people liked it and started contributing, then came branded T-Shirts, etc.
Now there's whole cottage industry in Boca with people spending weeks and months there, setting up and streaming from the cameras, they have trailers/control rooms, high quality equipment, daily and weekly updates, 24x7 streams, etc. NSF is a big player, Tim Dodd is another one, there's quite a few smaller players too.
NSF DOES seem to have some sort of agreement with SpaceX on streaming some of SpaceX's livestreams (ie when Ship goes out of visible range and SpaceX is the only place you can get video and of course telemetry). They didn't use to that until shortly after SpaceX streams moved to X (and immediately got replaced on YT by AI generated Elon peddling bitcoins).