It's often said on HU that this approach should be called "Healthy-Fat". Though the fats we eat should be healthy, I think "Low-Carb High-Fat" is the right name.
A name doesn't need to explain every tiny aspect of the plan, just enough to show what it is. Once you cut the carbs, you either go low calorie as well, or your need to up protein and/or fat. On keto and similar diets, we are replacing the calories missing from carbs with calories from fat; protein stays constant. There are low carb/low calorie diets (Fast800) and low carb/high protein diets (Total Wellbeing), but that's not what this group is for. Low-Carb High-Fat pretty well explains what diets belong here, and which don't.
It's very important to emphasise that it isn't high protein, because lots of anti-ketogenic articles emphasise the danger of high protein. I saw one yesterday, that even described the diet as moderate protein, but then went on to warn against kidney damage from a high protein ketogenic diet. 😒
Often lurking behind the "healthy-fat" statement is fat phobia. It's really hard after a lifetime of avoiding fat, to embrace it. Admitting it's a high fat diet is a way of reminding ourselves that it's OK to eat fat!
And it is a high fat diet. Even if you are not keto, I imagine most of us will be keeping below 100g of carbs. Protein is also personal, but I am for about 60g per day, so imagine some days it's as high as 100g. So that's a maximum of 200g or 800kcal total non-fat calories. My daily maintenance calories is about 2000kcal, which means I am getting at least 60% of my energy requirements from fat and on some days as much as 80%. That's a high fat diet by any definition, much higher than most people eat now and certainly higher fat than any of my ancestors routinely ate since the advent of agriculture.
With so much fat in the diet, you want to make sure it is healthy fats, but I don't think the label "healthy fat" is very informative to anyone not already in the know. People may envisaged the avocadoes, olives and maybe the eggs, but most people still think healthy fats are polyunsaturates, so saying healthy-fat does not describe the diet to an outsider at all. Even my well educated and corpulent foodie friend is was shocked that I was eating coconut oil; "that's the most unhealthy fat!" So, if we did call it healthy-fat, no one would be the wiser what we actually ate.
It is true that because we are keeping carbs as low as we can and don't want to eat too little protein, fat is the macronutrient that can be tweaked. If you want to cut calories, fat calories are the ones that can be cut. Interestingly, that doesn't change the nutrient mix, as the calories fat that is now missing from our diet will be replaced by fat from our adipose tissue. It's hard to cut fat, though. Replacing high fat foods with lower fat versions will also increase carbs and protein in the diet. Probably the best way is to reduce portion sizes of higher fat foods, listen to your body's satiety signals, and avoid those high fat foods that you find easy to overeat.
Wanting to try this! 
Skinny fat
Slowly getting into and used to LCHF. Coconut... Good or bad?! And how to work out how much Protein you should have a day?! Advice needed.