In the talk, I simply named most of the things after their function in my explanation and avoided formal terminology; however, if you have other interactions with this topic, it may be useful to know what other people call these.

The formal name for what I called "shift counter" in the talk – the little number at the top that said how many extra shifts (or zeros) the number had. See scientific notation for where the name comes from.
mantissa, significand
The formal name for the big number in the center of many slides (the display of the imaginary calculating machine).
base (of a number system)
Corresponds to the "shift factor": by what a number gets multiplied when you shift it to the left one digit.
A number system with base (shift factor) two. This is what most computers use, and was featured in parts II and III of the talk.
A number system with base (shift factor) ten. This is what most humans use, and was featured in part I of the talk.