Ok, now the story makes more sense.
So they say they were unhappy when they had played the first
vinyl test pressing (more likely meaning a test cut (acetate), not pressing), not a tape with the master of the (at that point) final mix.
That's an important detail because it shows that the problem was actually not that the mix itself was sonically bad pre-vinyl because of time pressure (as one could think from what Michael wrote in Moonwalk), but that indeed the runtime was too long for vinyl and they had to adjust things to work against the decrease of sound quality that naturally comes from vinyl. (There is always a bit of decrease compared to tape or digital, just with runtime that is that much too long (14 min?) it becomes very obvious.) It baffles me though that they didn't have this in mind from the start, and still attempted to put way too much runtime per side. Thriller was surely not the first album they had worked on, and back then vinyl was still the main release format.
That man states that if you have more music (measured in minutes) on each vinyl side, then this leads to less sonic quality.
That's correct. But you can't turn that logic around and say "the better the mix, the more space it needs".
The quality of the mix itself before vinyl and the state of sonic quality after vinyl are two things.
A sonically bad mix won't become better the more space is has on vinyl. It's just that the vinyl won't worsen the sound additionally.
So, one of the things they had to do was to mix again the songs (improving here their quality), which meant cutting minutes of playing time on each vinyl side.
The devil is in the detail again with this wording.
Cutting minutes out is editing, not mixing. "we're gonna edit them down", Bruce quotes Quincy in the video. That's the part that refers to making things shorter. Back then things were recorded on large multi-track tapes. To shorten things, these tapes were physically cut and glued back together, just like film was edited. You can't shorten songs by changing the mix.
Changing the mix can only be an additional thing. Vinyl (and especially with too long runtime) always also decreases things like the stereo panning. So depending on how exactly things were not sounding right on the test pressing / cut, they possibly also tried to adjust those things to have them come out better on vinyl.
Anyway, the info given is still very simplified, and so it's hard to really see what went on in detail. It's all a very long complecated process so there's still much room for speculation. For example the mastering also always plays an important role when preparing things for vinyl. Even the cutting engineer that prepares the master for the transfer to vinyl has room to worsen things by adding / not adding the wrong / right filters and limiters etc.