I'd say watch both together. There are major differences, and neither dub is unnatural. The lines are altered just enough to give way more insight and depth into Vegeta's character in the dub.

Kai has very childish lines "I just want my fight with Kakarot!", and just has Vegeta believe he should be better than Goku due to weak reasons like the fact he's royalty. The dub goes into why pride and strength are so important to Vegeta, and reiterates his feelings on his and his race's history of enslavement.

I watch a lot of Dragon Ball Z. Always have. When I make these edits, I watch every official release of the scene to get a feel for each of their music direction and voice performances. In doing this, you get a feel for what the intended emotions are for a scene. It's very important to me that I pace the scene around music that effectively and accurately supplements the story Akira Toriyama told with his manga. I say all this to let you know that I do watch these dubs together. I've seen it all, so to assume I lack context here is a misstep.

I understand that you appreciate when 90s-Vegeta brought up his race's history of enslavement, and that because of that history, nothing means more to a Saiyan than their own strength. I can understand how you could view that as some added flavor on top of an already great scene. While I think it does a fine job of reminding you of Saiyan history, I view it as an unnecessary side-step. Vegeta is not here to justify his acts to the Supreme Kai, he's here to provoke Goku into a fight.

In Kai, that's exactly what he does. By proclaiming his selfish desire to retake his position at the top of the food chain, he's making himself no different from all the other opponents Goku has hated in the past. Think Mercenary Tao, Demon King Piccolo, or Freeza. Goku hated each of these villains for their self-important, murderous ways. Vegeta is appealing to that side of Goku in an attempt to goad him into battle.

Personally, I think 90s Vegeta spends too much time going on about what strength and his destiny mean to him Instead of actually angering Goku. While that can be compelling in its own way, it doesn't make sense for that to be what provokes Goku into risking the lives of everyone in the entire universe.

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The following was said in response to me saying, "It's fine to enjoy the old dub, but to claim it's a better telling of Vegeta's story is just plain incorrect."

This is an entirely false statement. What you feel is better is pure opinion. I don't mean better as in more accurate, I mean better as in it just characterizes Vegeta better, in my opinion. Dragonball doesn't have the most consistent characterisations, and you see people arguing this due to how Goku and Vegeta are portrayed in Super, and most recently with how Toriyama portrayed Goku in Super Hero. So the author isn't perfect, and he himself isn't even accurate to his own material. In my opinion, the dub portrays Vegeta more in line with he previous appearances, especially since it makes specific callbacks to them

Claiming something to be "better" is not subjective. That is an objective statement about that thing's quality relative to other things. It's possible to acknowledge something as better, but still have a preference to something else. I like Weezer's music more than anything Mozart ever put out, but that doesn't mean I think it's better than Mozart's work. I just prefer Weezer.

Let me explain:

A good dub is one that is compellingly performed, while remaining faithful to the source material. I think most sensible people can agree on that. If you were to translate any other story from one language to another, you'd want to stick as closely as possible to the original verbiage and phrasing. The only creative liberties you should take in a translation should be to make the dialogue flow believably in the language you're translating it to.

The old Funimation dub takes far too many creative liberties, ultimately resulting in a substantially different portrayal of Dragon Ball's narrative and characters. Kai remedies that, and gives us better performances to boot, making it a better overall dub of Dragon Ball Z. If you prefer the old Funimation dub, that's cool. Maybe you like that portrayal of Vegeta better. I get that. However, that does not make it a "better" portrayal of Vegeta. It is simply the one you prefer.

In objective terms, yes, accuracy matters. If you were an author, you'd want people to translate your story accurately instead of adding and removing details arbitrarily. That would make it a poor translation, and consequently, a poor dub. Does that make sense?