nope, not the same, I agree.
A mobius is pretty much knit outward in both (?) directions. There will never be a cast on edge. (Truely it is only one direction, but it comes out to a flat piece of knitting)
If you have a cast on edge that stays where it is: No Mobius!
if the cast on is in the middle and there is no end to what you are doing: Mobius.
I yet have to make my first, but I have memorized it before - and know the Mobius band since childhood. The trick is the half twist.
you can try it with a strip of paper:
take a strip, a few inches long, give one end a half twist, glue together with the other end. make a line of pen on one side and see, where it ends: it goes around the outside, goes to the inside, and back out, and it still is all flat and twisted just the same.
If you would now cut it back open, there would be a line on either side of that (now flat) strip.
If you have given the paper a FULL twist: only one side would have a line after this procedure and the line would be half as long (with the same size paper strip).