I read a couple of your quotes and I am not sure where you are getting your info from from wc3 or if you skipped bunch of parts while playing/misinterpreted some things? To clear few things up for you:
Firstly, even the undead npcs/cultists have quotes like "My life for nerzhul and "for the lich king" so it is pretty obvious that he existed and was referred by both of the names. If you got too used to the Arthas model - google how LK model looked like stuck in ice in wc3 and you can see why he was called the "lich" king
Lk description existed in game manual and I still have it in my old room in my parents house I think
It was pretty detailed description of him and made it clear that it was Nerzhul.
In game, there is no references to LK that makes him look "ancient". He is mysterious for sure but that doesn't mean ancient. What were ancient in the expansion were the nerubians which are not directly related to LK, just one of the races that LK dominated after establishing in Icecrown. Anubarak/underground missions were there to provide backstory for anubarak and the nerubian kingdoms as well as an introduction/easter egg for the old gods with the "faceless one". (For the people who thought the ulduar/old god patch felt out of the blue in wotlk - the roots of the old god in northrend goes all the way back to wc3)
On top of that, in the beginning of the wc3 we are sent to investigate "the new plague" and then encounter the new mystery cult so it was made pretty obvious that this was something new.
+ Tich explains how frostmourne ended up in Arthas's hands and goes into the story of LK's origin/purpose as well within the story missions.
+There is a whole exposition "walk" that arthas takes with KT before one of the missions where KT explains more of LK's backstory and origions, as well as, motivations.
P.S - Contrary to popular belief, blizz's writing was never "good" and had a lot of holes in the story + they always preferred rule of cool over logic/consistency. However, the origins/backstory of LK is not one of these problems and is actually quite fleshed out and consistent for the most part (until upcoming shadowlands BS). There are major major problems with the LK story tho, for example:
1) the "ice" that LK was originally trapped in wasn't some random ice, it was a twisting nether crystal which KJ put him in to enhance and amplify his telepathic powers to a godly degree that he had control over all the undead on azeroth through it. However, when Arthas shattered it and put the helm on, he should have lost this power and only had control over undead on a limited proximity since he no longer had nether crystal powering him.
1.2) why the fuck did bolvar trapped himself in a random piece of ice after the LK fight? I understand they wanted him to look like OG LK, but knowing the above its completely pointless/the rule of cool only.
2)also, if his plan was to merge with arthas, what was stopping him from merging with one of his own powerful undead commanders as a temporary body beforehand, since we see that bolvar took the helm after arthas no problem, there could have been someone before arthas as well and LK could have even met him half way!
3) We learn early on that LK claimed the soul of arthas as soon as he grabbed frostmourne in the game. how come after LK commanded arthas to make his way to the throne to "merge" with him, nerzhul didn't have full control. We know from the novel that it was arthas that took control somehow? although his soul should have been trapped in frostmourne and under LKs control.
4) there are bunch more but ill /endrant.