Maybe you said this before but when you get the NISL you are looking at the right edge of the cb and with your peripheral vision you are seeing a center cue ball and extending that line through the center core of the cb and out to the ob, right? At this point your head is angled toward the target pocket and your eyes are focused on the right cb edge. When you have the NISL in your peripheral vision do you then shift your eyes and head directly over that line as you put the cue over that line, or do you maintain an offset? If so, then it seems you are moving the cue askew from where your vision is. I'm not successful in seeing anything different when looking at the right cb edge but want to be sure I do what you are doing.
I think I can clear this up. Firstly, my eyes stay at the offset. As I've already stated, at any point I can look back at the AL and SL if I choose to. Second, "CCB" and "NISL" are synonyms. We didn't have "NISL" terminology until the book. It was always stated as "CCB is the target", which is for all intents and purposes, the NISL.
Now from this offset, there are two cueball centers, determined by which eye is dominant for looking at CCB. They are extremely close to each other, within millimeters, and each on either side of the SL. Two NISLs, two shot lines, two tracks to two separate pockets. We can focus on either CCB/NISL. To the untrained eye this isn't a simple feat to focus on the either center at will. So we have two ways to lead us to the intended CCB.
The original method is using a strict 1/2 tip pivot. If you place the cue stick 1/2 tip parallel to the RIGHT of the SL, and pivot your cue to CCB, you will be pivoting to the CCB for right pivots. If you place the cue stick 1/2 tip parallel to the LEFT of the SL, and pivot your cue to CCB, you will be pivoting to the CCB for left pivots. An important detail here is that
you are pivoting to a CCB that you can already see.
It is not true that the CCB/NISL is determined by the strict pivoting motion of the cue stick. That said, I think that is exactly how it was always taught, and TBH if you are careful about the 1/2 tip offset, I'd say the strict pivot is going to be extremely near, if not dead on every time. But it is not out of the realm of possibility that you might make a very slight adjustment to the bridge hand to put the cue on the CCB/NISL. "eyes lead, body follows". All the same result.
Now the new method, I which I believe to be superior to manual pivots, is stepping the CB. That is, focus on the right edge of the CB to see the right CCB, and focus on the left edge of the CB to see left CCB. This is just a way to force your eye focus onto the correct center. Once you have the correct focus, you can move your cue into place and then switch your eye focus to CCB/NISL and dial in the cue stick. In this case, the exact pre-pivot position of the cue stick is not as crucial. This is evident in my videos, I've used stepping for so long I don't really pay close attention to moving into an exact 1/2 tip offset, although as a force of habit I'm probably very close.
So I'll restate an important detail again. All of the steps of CTE can be done with only the eyes (no cue stick) from both ball address and full stance (I noticed Dan saying ball address in his video when I was in full stance, so be careful about that too.) So for instance. At ball address (or "out of the box") I can put my eyes at the offset to see the AL and SL, then step the CB to see the NISL. No cue stick involved. Now, I can move into full stance (or "in the box") following the SL. My eyes are still at the offset where I can see the AL and SL, and again I can step the CB to see the NISL. No cue stick involved. It is at this point I move my cue stick onto the NISL. This can be done by a 1/2 tip pivot, or if you prefer, once you get used to CTE, you can just move your cue right onto the NISL in one movement as you come into full stance. That is typically what I do when not in demonstration mode.
The unique perception for every CB/OB orientation starts at full stance. My physical alignment to AL/SL is unique for every orientation. If you are looking to unravel the "magic" around CTE perceptions, that is where you need to be looking.