This movie, done in the live action / animation style of Thief's cutscenes, shows some of the events that would happen in typical gameplay. This movie does not appear in the game, it's taken from the Thief non-interactive demo on the Eidos Demo CD.
A nice long clip showing what gameplay in Thief looked like in '96. Yes, this is very, very old. Looks to be talking place in an early draft of the Lord Bafford mission.
Garrett decides to go in and steal it himself: because of their internal arguments, they won't blame him. He intends to break in through a gambling den they operate, the Overlord's Fancy.