The 4th Doctor clip is of him on Galifrey during the episode "The Invasion of Time" part 2 of 6. The Doctor walks down a hallway and it is Leela not Clara who is pursuing him. The Sontaran invaders don't make an appearance till the very end of episode 4. So there is no immediate threat Clara is saving him from in episode 2. Also it was the Doctor's plan I seem to recall to let the Sontarans penetrate Galifrey's defenses and occupy it.
My understanding of it was that she wasn't involved in what happened in the episode shown but that the GI instituted something new which she deflected/diverted/stopped, often without the Doctor even knowing there was anything going on.
The episode of Doctor Who went on exactly the same with or without the Great Intelligence or Clara, but there's an "Adventures of Clara" episode going on at the same time.