The IIIE did indeed have a Centaur stage with a "thermos" full of liquid hydrogen and another with liquid oxygen, but that's not what we see in this clip -- the pillars of fire and smoke come from a pair of solid-fuel boosters that burn for about two minutes, followed by about six more minutes of flight powered by two more stages burning non-cryogenic liquid propellants (hydrazine and nitrogen tetroxide) before the Centaur was ignited.
About 15 years earlier, I had done some work at the VAB and walked around the Saturn IV that was laying on its side there, just as James Burke had a few years earlier. It wasn't there when I worked the above referenced mission. I'm not sure when they moved it, or where it ended up (but it's not in the "rocket garden" at the Visitor Center).
If you mean just the Saturn IV (no B), that was the second stage of some early Saturn rocket. I don't know much about those or if one was ever on display.
Centaur has balloon tanks which is just very thin metal. It can't even stand on its own without pressure. But no thermos.
That said, on the Titan IIIE at launch, there was an second outer surface around the Centaur -- the payload fairing -- which was discarded only after atmospheric pressure was low enough that it was no longer needed to protect the payload and the Centaur upper stage. (It wasn't a true thermos as the space in between wasn't sealed and was at ambient pressure at launch).
See the cutaway views on pages 16 and 17 (pdf pages 19-20) of https://www1.grc.nasa.gov/wp-content/uploads/TC-6-Voyager-Fl...
You're correct about payload fairing. Also, Space Shuttle External Tank was spray foamed, causing the Columbia disaster when a piece of foam from the tank with hit the wing leading edge. I was googling for what the final verdict was but couldn't easily find an answer. Nice summary in [1]. Cryogenics are complicated.
1:https://idlewords.com/2003/08/things_i_have_learned_about_fo...
So knowing, the engines starts 3 seconds before liftoff, he was able to time the speech.
I saw this clip dozen of times, and only after keeping attention I realized this.
So with the cut technique, it seems like seamless feature, and that's the magic of the TV. Nevertheless amazing delivery.