The only reason it's confusing is that you insist on keeping yourself confused. Very few people are going to fall neatly and entirely into a single category - even one as broad as liberal or conservative. These words don't exist so much as to define people as to define points of view. It's only when a person's ideology is dominantly characterized by a particularly category of viewpoint that we say that he's a [insert whatever noun applicable]. If you don't approach it with that attitude, then everything is going to keep vacillating between "vacuous" and "redundant", and you wouldn't be able to make sense out of anything in life.
So as regards the Cold War, there was a significant neoconservative element to Republican thought, which should come as no surprise, as it was the the original (Xxxxxxxtz) Neoconservatives who did a lot of the heavy lifting to make that a reality.
Yeah that's probably it. Couldn't be anything else.
Sarcasm aside, in case you sincerely don't understand my logical problem with what you've been trying to say, here it is.
You claim "neocons" are people who advocate stuff like X, Y, Z for reason A.
When pressed for examples of "neocons", you tend to say stuff like "well take those people, Them, who advocated Y." Thus conveniently forgetting the "for reason A" part of the argument.
And when I ask you "but what if they were advocating X for reason B", you're silent or change the subject a little.
There's a loose end in there that you continually fail to tie up, which results in your "neocon" definition being either vacuous (because you never quite demonstrate the existence of anyone who advocates X, Y, Z for reason A) or straw-man (because you implicitly deny the logical possibility of advocating X, Y, Z for reason B). It's never both vacuous and straw-man at the same time, of course, because you vacillate between the two constantly, when the discussion demands it.
That's about the long and short of why your spirited defense of the "neocon" term confuses, and I don't know how else to explain it, and don't really want to write any more opuses about it. But frankly, I shouldn't have to, because I believe that you're more than smart enough to see what I'm saying.
You just really, really need the word "neocon". For some reason.