So, if you go out and buy
VC or
TK army book you'd use that book as is. Slain general/ hieriphant included. GW did this on purpose so their is no confusion, in theory.
When you buy the Nagash book and play undead legions, the two pages everyone is referring to is the undead legion's army special rules. It flat out says the undead legion is it's own army with its own rules and only uses the unit entries of the other book, and use the new army special rules here in the Nagash book. The two designer notes are in context of the undead legion army special rules pages, one being nehekharan undead = undead and slain general/heriophant is intentionally omitted. No where does it say these rules affect the original army books, though slain general/hiero fluff wise makes since as the Nagash's influence could be loosely interpreted to mean all undead armies. RAW is not though, it's referring to undead legion armies only. Likewise, if you run a
TK army out of its book, you don't errata the special rules or the nehekharan undead rule. Though I wouldn't blame you wanting to
And, as far as Lore of Undeath, any wizard that has a lore they can choose or come with gets Undeath as a choice too. When they summon its from undead legions units (not goblins and orcs or bretonnians) and they follow undead legion army rules. This means those units ignore your general dying.
It also means a vampire counts or tomb kings army out of its book could run undeath, summon undead legion units, have it's general die and it would make crumble tests for its own units but not those summoned through the lore.
But really, most undead players should be running undead legions. So unless you want special characters not available in the legion, there is no reason not to. So all of this is sorta a mute point.
Side note: armies of the end times rules are not attached to undead legion or lore of undeath. Each of these is a seperate piece of a larger whole.
With all that, I personally think 9th edition undead is going to follow rules similar to this. They are trying to make them function as allies easier and losing the whole crumble when general does is part of that.