Best part about those is that you can actually reseal them effectively, unlike a cork.
My family includes winemakers. As far as I know, the taste doesn’t have a special name. And it’s correct that it’s mainly to test whether the wine is spoiled is some way, either by the cork failing or one of a few other ways. That’s why even a twist off bottle should be presented for a taste. A sniff may be all that is needed, but a taste gives you a bit more information.
A friend of ours loves wine and has a real wine cellar. A few years ago he gave us a bottle of his favorite French wine. I was saving it for a special occasion, but he was anxious to hear my opinion and kept telling me to just drink it.
So, I took it with us to a little Italian restaurant that was BYOB. The owner grabbed a corkscrew and began to open the bottle, but the cork crumbled. She eventually opened the bottle, but a lot of the cork was in it. She poured it through a coffee filter, and we all shared the wine. It was good!
I did a search and discovered I could buy another bottle for $450. Yikes!!
This could have been because the bottle wasn’t stored properly (on it’s side in a cool place) but as long as it did it’s job the wine shouldn’t be affected.
This How To article says “sample the wine” without giving the ritual a specific name.
As does this one.
You’d think the French would have come up with a term for the ritual
Oh, the wine was delicious! But it was a farce, dumping coffee grounds out, putting a new filter in, and straining the wine. Then later discovering it was an insanely expensive (to me) wine was the pièce de résistance
Reminds me, fondly, of the time my brother drank Maddog 20-20.
Ah, the lovely grape stench…uh…aroma the carpet had, for what seemed like years. It was vintage by them.
Don’t ask him if he wants a little taste of it now.
Did he puke it up, or spill it?
… not that there’s a whole lotta difference, I guess.
Puked, barfed, rolled around in. Heaved some more.
Next morning he was decidedly green. And, had to go puke some more when he came into the living room where the carpet was, with a big purple stain.
What I love about that scene is that the cork malfunction wasn’t scripted, it was a prop failure and half of that scene was improvised. It’s one of those accidents that was just meant to be.
ETA: LOL, I just noticed that same point was made in the first comment on that very video on YouTube. 
True, but there are all kinds of wine bottle stoppers you can buy that do a fine job. If you’re really picky you need to get one of those nitrogen kits the prevent the wine from oxidizing by displacing all the air out of the bottle.
The stoppers I use have little plastic levers that you pull down, causing the rubber seal to expand and become completely airtight. Modern screwtops like Stelvin do a fine job and in many ways are superior to corks, and allow the wine bottles to be stored upright which is often more convenient. But I like the tradition of corks, and the fact that you can tell a lot about a wine by examining and sniffing the cork, which should always be presented by the sommelier along with the tasting. A cork that is moist and deeply red shows that the wine has been stored properly, and its bouquet tells you a lot about the promise of what you’re about to consume.
A friend once gave me a wonderful gift which was a wine aerator, to accelerate the breathing of red wine. It’s a cleverly designed acrylic gadget with a tiny hole that somehow sucks air in as the wine pours through it. And it comes with a fine-mesh screen mainly intended for filtering out sediment, but it filters out cork bits, too. You could have used one of those! Breathing and filtering all in one step!
I find that a blender does the trick amazingly.
Hyperdecanting wine à la “Modernist Cuisine” – Stefan’s Gourmet Blog (stefangourmet.com)
There’s a French word for it; goûter.
It means “to taste” but it’s French so therefore fancier.
The term dégustation also translates as tasting, which is more the entire act of observing the wine’s qualities and judging it’s taste. I suppose this term is more the “ceremony” being described.
Attack?
What are we attacking?
There are multiple phases to formal evaluation as initially developed by French wine connoisseurs: the visual examination, the scent, the initial taste, the lingering impression, and so on. “Attack” (or originally “attaque” in French) is the word that describes the immediate “hit” the wine gives you when you first taste. This is applicable for all tasting activity, and has no specific relevance for the corking check being discussed in this thread.
I think the OP is about the tasting of the first small sip offered. (The title) Not necessarily corking.
Is the ritual perhaps called corking ?
I read that the attack was part of the drinking of the wine. I just thought maybe that’s what they were looking for.
The tasting of the first small sip is specifically for the purpose of checking that the bottle has not been spoiled by corking or some other contamination. It is not a tasting to see whether or not you “like” what’s in the bottle, though it’s true that most restaurant-goers don’t understand this and have no idea why they’re being offered that little sample. The point is, “corking” and that first taste are inextricably intertwined; the OP explicitly says that the little splash is provided to “make sure the bottle is not corked and undrinkable.”
“Attack” is a term that refers to the process of tasting for the purpose of evaluating the actual qualities of the wine. It is not related to the “see if the bottle is ruined” taste.
I’ve had it happen twice but over lots of bottles of wine over 30+ years it’s really very rare to happen.
Also, don’t sniff the cork. It just smells like wine and cork. Just smell the wine and, if you want, inspect the cork to see if it is cracked or crumbly which may suggest the bottle has gone bad.
I taste the wine (after smelling it) because why not?
Ah, got it.
So at a wine tasting thingy how much wine before you can describe the wine with all those nice clever words?
One drink, Two?
Or do you smell, sip, spit and know?
How many different wines before you can’t tell which is which?