In Knitting Without Tears
by Elizabeth Zimmermann she talks about a loop cast on page 20. It is pictured there and is more like the M1T that Amy shows in her increases. Then she adds, "If you can train yourself to twist the loop
(EZ italics) before putting it on the needle, it is a great improvement, and may perhaps find yourself using this version frequently."
I found that the picture of that technique which she called Twisted Loop Casting-on is not made by doing the M1T though, but a M1A (which is to fold the yarn back on itself so the the yarn going to the skein is in back). Then with that e-loop formed, instead of putting it on the needle the most direct way(like the M1A), to twist it toward the RH needle tip and when the next opportunity comes to put in on the RH needle tip, put it on. If you try to twist it the wrong direction it will all come apart, so it is self correcting.
I think this is what your directions are talking about. I can probably get DH to take a picture of how this looks if anyone needs it.