I’m bummed, because I listened to the two selections, and could only hear the obvious differences: number of singers, softness of performance, and so on. I couldn’t perceive what you are describing.
If what I linked to is not vibrato…what is it? Tremolo?
The number of singers is the same is both videos - 4 in that particular passage. Softness… just adjust the volume. It also helps if you’re using good quality headphones or speakers. 
Here’s a video with very clear examples and discussions about what vibrato is:
I get a message saying “This video is not available”.
I thought it might be restricted by country, so I tried using a VPN from both the US and the UK, but I still get a message saying it’s unavailable.
Is it a private video? Or has it just been deleted?
The link still works for me. You could also try this one or this one, recordings from the same album (Gustav Leonhardt Box set), also sung from deep in the maxillary antrums and sub-uvula.
That’s what I get for indiscriminately downloading monster Baroque albums on the cheap. 
I was illustrating this singing style for Mrs. J. yesterday, and the song I picked to do it was Blondie’s “Call Me”. It worked eerily well.
Roooll me in de-signer sheets
I just cawnt get e-nough
No, I still get “This video is not available”.
It may have something to do with the fact that it’s copyrighted content, and YouTube wants to display ads. But turning off Adbock Plus and Noscript doesn’t help. I don’t really have time to fiddle with this unfortunately. If someone else has a solution, please let us know.
Grin! What I need is a pair of higher-quality ears! (For me, “high-end audio” only means anything with more signal than noise!)
Thank’ee for that link! That made it clear!
I guess, personally, I’d side more on the non-vibrato side of the debate. It sounds more “pure” to me. On the other hand, I really like the movement in the clip I linked to, the aria from Bach’s Hunting Cantata.
I had asked about tremolo, but find that that’s something else entirely. So, what is the term for the vocal leaps/flutters/variations in that aria? I’m sure it has a name; in music everything has a name!
One more from a different-looking source.
The singing starts around 1:05.
Okay that works for me!
This is a recording of Alfred Deller, who was a famous countertenor, or male alto. So are you referring to male altos singing falsetto?
This is different from the high-larynxed singing mentioned in the original post, and it doesn’t cause any harmful effect, or damage the voice. Many pieces of Baroque music were written with alto parts, which were originally sung by male altos, so this is completely authentic.
Here’s a 1992 documentary about King’s College Choir which has some discussion about male altos, with some examples of alto singing. Watch from 26:44 for about 2 minutes.
I wouldn’t have guessed that was a male singer, but no, falsetto is not the problem.
It’s the antrums thing.
Ah! Alfred Deller – and his son Mark. Wonderful singers; I have a number of their recordings.
I’m trying to understand it, myself. Is this the sound? Kind of a just above the crack in the voice Mickey Mouse tone?
Here’s a kind of parody of it.
I haven’t listened to the pieces in question, but I wonder if you might be referring to pieces written for castrati. In modern performance, these parts are sometimes sung by a countertenor, but may alternately be sung by an alto. In at least one piece for which I was a chorister (it’s been so long that I don’t recall which one), the alto soloist sang a particular section with a distinctly flat, sans-vibrato tone that really stood out. It was my understanding at the time that she was filling in (so to speak) for the castrato.
Wowsers! Thank’ee! That’s what I was thinking of…but even moreso! The second example, in particular, totally whipsaws the notes! I’m impressed!
And, aye, grin! Vocal acrobatics for darn sure!