I think he does explain it the way you saw it. At least on page 43. I haven't re-watched the video.
You start with an Am chord. If you call it a vi (Aeolian in C) the F and G are the IV and V, but D Major doesn't fit because it would be minor in C Major.
If you then consider the Am to be a ii (Dorian in G), then D Major would be the V.
So first you treat the Am a vi in C Major (Aeolian), then switch to treating it as a ii in G Major (Dorian) when the D occurs .
Then he states that this switch from C to G (A Aeolian to A Dorian) over the D chord is only temporary and goes back to C Major (Aeolian) on the F chord.
It's simpler for me to treat the Key as A minor and start with the A minor scale. (A B C D E F G)
Raising the F to F# to accommodate the D Major chord changes the scale to Dorian. (A B C D E F# G)
So it's just a matter of changing, and accenting, that one note whenever that D chord comes along.
Either way of thinking about it works out the same.
In Changing Modes 2, you start out in F# Dorian (F#m and B7= ii and V7 in E Major).
Over the D & E you change to A Major. (D & E are the IV &V.)
However, A Major = F# minor = F# Aeolian.
So switching from the Key of E to A making F# the tonal center = F# Dorian to F# Aeolian, the only difference being the D# in Dorian vs the D in Aeolian.
Notice there isn't actually a Key Signature in either example, i.e. no Key Signature doesn't always mean the Key of C. Depending on the publisher, sometimes you gotta figure out modes for yourself.