You don’t remember wrong.
In another video, he says that the switch mode wouldn’t work against units like Rams giving the example of the Samurai.
That being said, there was also a multi purpose cavalry for the Persians with the same mechanic, which probably got morphed into the Byzantine Cataphract (which as you say was also multipurpose).
If Cataphracts were regional units, Byzantines and Persians would surely get them.
Probably Chinese as well.
So I get how the Persian UU became the Byzantine UU.
Edit: it is also possible that this mechanic was originally planned for both units, considering the Cataphract is a mounted unit and the Samurai is a foot soldier.