A word of caution here-- what the Pros dance, and what may be required as an answer in a prof. exam-- possibly may dis agree . -

As judges, we do allow " poetic" licence on occasion .

How do you teach it.Is it a step to the side as it is written. What does Marcus Hilton do.

It depends on the individual or the class-- if its medal class work-- book adherence is necessary-- i do ,-- sometimes -- show alternatives .

Remember-- the book is a guide, and there has been some disagreement over points of techn. for many yrs ( classic e.g, till changed -- 4th step nat. going into spin turn )
Evolution in dance, is ongoing-- with music changes-- style changes have occured-- the " Modern " trend .-- samething happened when the 2/4 signature changed to 4/4.

At the Youtube video he underturns 1-3, step 3 backing DW and does a 1/8 between 3 and 4. I don't understand why he does it that way, I would do it as written in ISTD technique.

Step 2 I dance like written in the book "to side, backing DW" and this means NOT going along LOD but following the line DC. In some medal classes for making an easier sequence I teach the Rev. Turn starting facing LOD and doing 1/8 more between 1-2. Even in that situation man doesn't dance step 2 along LOD (like many beginners want), but should dance 3/8 to left between 1-2 to get step 2 moving DC. This step moving along LOD would result in big trouble concerning lady's heel turn, especially her following step 3, because man would drift away instead of dancing AROUND the lady's hell turn spot. With beginners making that drifting-away mistake you even often see the lady stepping OUTSIDE at 3, not able to follow the man.

German Teacher. I would draw attention to Alex Moores book. A Reverse Turn. When it says on step two, RF to the side across LOD continue to turn on ball of RF and step LF back. That is exactly what it means. When it says that between 1 and 3 there is three-eighths of a turn doesn't alter the above alignment in any way
Among the general notes it says . Do not let the third step swing outwards. Keep it well under the body. Which is exactly how Marcus Hilton
Chris Hawkins Timothy Howson do it.
At no time is it written that any of those steps, especially step two, will have you backing diagnal to the wall
Which of the books say to the side backing diagnal to the wall. It's not in mine.

I said step 2 moves DC (ISTD "Ballroom Technique": step 1 facing DC, step 2 backing DW with 1/4 between 1,2) - BTW moving DC is moving across LOD. That topic is NOT a problem of the Hilton video, because he uses a complete different alignment.
I see at the Youtube video man's step 3 ending DW instead of LOD - then he does an 1/8 between 3-4 which is quite strange because it all breaks the line of transfer energy into the next swing of the Feather Finish.

German Teacher. This is the way i would work it out. I turn at the end of step one. If three is straight back down LOD, feet brushing. Then where does step two go.
The teacher who introduced backing diagnal to the wall on that second step some fifteen years ago. Today doesn't teach that way. If he does nobody is listening. We all have eyes so lets look for ourselves at what the best dancers in the world are doing.
If we are doing the Reverse Turn where the lady passes the feet and doesn't do a Heel Turn and the steps are O/S partner, thats a different story. The second step will be a backward step and the ladies steps are forward.

"Which of the books say to the side backing diagonal to the wall. It's not in mine."

All the books used by the major dance associations for their teacher examinations!

ISTD Ballroom Technique, used by many associations.

Guy Howard's Technique of Ballroom Dancing, The IDTA's bible.

The UKA Ballroom Book, by Ken Akrill (P)2006

If you look at the footprint diagram in Alex Moore you will see that step 2 is clearly placed backing DW. The RF continues to turn as step 3 is taken.

Social Dancer. On the chart , where are the feet after step two is completed. If at the competion of step two you are backing diagnal to the wall in which direction would you be going on your next step three.
Now cross over to the ladies steps. Does that corespond to your version of the man's steps. Wouldn't it be peculilar with the lady on one alignment and the man on another.
But lets forget all of that and you tell me if you Goggle Marcus Hilton youtube Basic Foxtrot. Tell me exactly what you see on step two of a Reverse Turn in the Foxtrot and is step three straight back down the LOD, are the feet brushing at that point, are yours. Ask yourself . Are you at the end of step two perfectly balanced. Or do you have to catch yourself on step three which in that case would not be a perfectly balanced step to the rear straight down the LOD.
To finish, I've heard all of this argued out by much cleverer persons than you and I many times before.
If you want an example a bit closer to home. Go to Learn the Dances on this site and select Reverse Turn Foxtrot. Tell me what you see. Are you converted.

@ Serendipidy:
What do you mean about different alignments of men and lady? Man is backing DW e/o step 2 - lady facing DW, she does an 1/8 between 2-3 facing LOD and he does also 1/8 between 2-3 backing LOD. In the learning center I see nothing else at the video. And now again my favorite priority: Where is the lady's spot of rotation at her heel turn? - E/o step 1! That means the man must NOT MOVE AWAY from that point until e/o step 2, so step 2 cant't move along LOD but has to go along a line DC, so he moves AROUND the lady. This you see at the old Alex Moore diagrams as well as in the actual ISTD charts given as "side, backing DW" (like you correctly said "e/o" the step). A step moving along LOD with the preceeding step facing DC would have been given as "side, slightly back" or "diagonally back".