Special term for ski boots. Hiking boots, riding boots, wellingtons, army boots, all have 'high uppers', but ski boots have cuffs.

"Some of the parts, features, and properties ski boots can have are listed below.
Shell
The shell is the solid outer layer of the ski boot, and is made of two parts, the lower shell and the CUFF. The lower shell is the part where your foot is contained, and THE CUFF IS THE PART THAT GOES AROUND YOUR SHIN AND LOWER LEG. [ ... ] some ski boots, generally boots aimed at lower intermediate skiers, have a mode that makes it easier to walk in the boots. Exactly what these modes do changes between different boots, but they normally free up the CUFF to rotate forwards more easily, and often let the cuff come further backwards than normal [ ... ] CUFF ALIGNMENT (sometimes refered to as canting) allows the sideways angle of the CUFF to be adjusted to suit the skier's lower leg shape. For people with a knock kneed stance the cuff needs to be tilted inwards, and for people with a bow legged stance THE CUFF needs to be tilted outwards. Most boots offer about 1 degree of movement in each direction. Adjusting this allows the boot sole (and therefore the skis) to sit completely flat on the ground in your neutral stance, as it should be. CUFF ALIGNMENT is normally adjusted through bolts on the outside of the ankles."
https://www.mechanicsofsport.com/skiing/equipment/boots.html

"At the elite level, every small tweak to a ski, boot, or binding setup can be measured, felt, and appreciated. In the quest for speed and performance, all options are explored and tested. What to do with BOOT CUFFS and CUFF ALIGNMENT is one of the small steps that is often misunderstood, ignored, or improperly adjusted, but it has a direct correlation to overall alignment of the leg and boot and will dictate a ski’s response on snow."
https://skiracing.com/your-straightforward-guide-to-boot-cuf...

Tha B-A BA of putting on ski boots:
"1. While wearing proper ski socks, insert your foot into the boot―just like Cinderella.
2. Start with the LOWER CUFF BUCKLE first.
3. Once it’s secured, put forward pressure on your shins to flex the boot forward. This will allow your heel to slide back into the correct position and take pressure off your toes.
4. Fasten the UPPER CUFF BUCKLE and secure the gigantic strap at the top.
5. Now that the ENTIRE CUFF IS LOCKED IN, flex forward again to make sure that the heel is right where it should be.
6. Give the tongue an upward tug to situate your shins and ankle.
7. The very last step is to fasten the toe buckles.
https://skiisandbiikes.com/blogs/ski-guides/how-to-buckle-yo...

"CABRIO : A type of ski boot constructed with three parts: lower shell, CUFF, and tongue. Usually has three buckles and a smoother flex compared to overlap boots.
OVERLAP: A style of ski boot shell that has a lower shell and CUFF that wrap all the way around the foot/leg and overlap, especially in the CUFF. Usually features four buckles."
https://www.skimag.com/gear/ski-and-boot-terminology/

CUFF: The CUFF is the removable UPPER PORTION OF THE SHELL, attached at hinge points on either side of the ankle.
https://www.skimag.com/gear/anatomy-ski-boot/



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Note added at 1 hr (2022-11-18 14:46:55 GMT)
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Note that in French too there is a specific term for "cuff" : "collier".

So the question is, do you use the layman's term (to the extent 'tige haute' is a layman's term) or the sports jargon? Is it reasonable to think that someone buying ski boots knows the jargon?

"La coque :
Se compose de 2 parties, le sabot et le COLLIER. Elle est moulée dans des matériaux plastiques plus ou moins rigides selon le niveau de performance recherché, transparents ou non, dont l’objectif est aussi de nous protéger de l’entrée de la neige dans la chaussure. [ ... ]
LE COLLIER :
Enveloppe la cheville et le bas de la jambe. Fixé sur le sabot au niveau des malléoles, il est articulé pour permettre la flexion des chevilles et des genoux, et se ferme en portefeuille sur la plupart des modèles. "
https://snowflike.com/conseils-bien-choisir-equipement/conse...

"Le COLLIER, LA PARTIE HAUTE DE LA CHAUSSURE, est différent entre modèles homme ou femme afin de s'adapter aux différences morphologiques (insertion et forme du mollet). Il doit donc maintenir votre cheville et votre tibia sans points douloureux."
https://www.SALOMON.com/fr-fr/stories-guides/skiing/advice-w...


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Note added at 1 hr (2022-11-18 14:54:14 GMT)
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I'm not sure that 'tige haute' is entirely appropriate, since in other forms of 'tige haute' footwear the upper part of the 'tige' (which despite its name has nothing to do with verticality) enclosing the ankle and in many cases above is continuous with the part that encloses the foot. In ski boots - modern plastic ones anyway - it is separate, so is not really part of the 'tige' at all.

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Note added at 9 hrs (2022-11-18 23:01:00 GMT)
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Note that cross-country ski boots come in two (at least) styles, with high or low cuffs (tiges hautes, tiges basses). I don't know that for downhill skiing, as is your case, such a distinction is made.

"Choisir les bonnes chaussures de SKI DE FOND
La chaussure que vous choisirez dépend de la discipline que vous voudrez pratiquer :
• Skating : il vous faudra en revanche des CHAUSSURES DE SKI À TIGE HAUTE DOTÉES D’UN COLLIER pour un parfait maintien et une bonne impulsion.
• Classique : dirigez-vous vers des CHAUSSURES À TIGE BASSE pour que vos tibias puissent être libres de leurs mouvements.
[...]
chaussures de SKI ALPIN [...]
Freestyle : ici, le COLLIER est plutôt droit pour un positionnement du corps neutre en entrée de module. Le flex d'une chaussure de ski freestyle est assez rigide et on retrouve des amortisseurs sous le pied pour améliorer le confort en réception. [...]
Racing : pour vous accompagner dans la performance en slalom comme en géant, ces chaussures de ski racing disposent d’un COLLIER très penché vers l’avant. Le tibia est ainsi parfaitement positionné sur la languette. Elles sont munies de 4 crochets pour un maintien à toute épreuve, et elles sont plutôt étroites pour un transfert de puissance maximal."
https://www.achat-ski.com/6-1024-chaussures-de-ski.html

"Petite note concernant les CHAUSSURES SKI DE FOND : vous en trouverez de 2 sortes. Les CLASSIQUES OU « À TIGE BASSE », sont souples pour bien dérouler son pied. Et celles POUR LE SKATING, qui sont PLUS RIGIDES À « TIGE HAUTE » pour un bon maintien de la cheville et une meilleure adhérence sur la neige"
https://www.ski2alpes.fr/conseil-etre-confortable-dans-les-c...

"SKATE SKIING BOOTS
Skate skiing boots have a rigid sole that reduces flex – as well as ADDITIONAL SUPPORT IN THE CUFF – that provides better leverage and less twisting throughout the skate skiing stride. Entry-level boots use plastic in the CUFF whereas high-end models feature carbon CUFFS that are lighter and provide increased stiffness. "
https://www.salomon.com/en-us/stories-guides/cross-country-s...

"CLASSIC CROSS-COUNTRY SKI BOOTS have a soft outsole which promotes forefoot flexion. They have LOWER CUFFS for a greater range of movement and are generally looser around the ankle.
SKATE BOOTS generally have a more rigid outsole which helps you to lift your ski up and reduces the possibility of torsional movement. The HIGH CUFF and spoiler help keep your ankle in place and provide lateral support."
https://www.salomon.com/en-us/nordic/nordic-advice/cross-cou...