Dieting is calorie counting in some way or the other. Eg: Weight watchers uses a point system. It’s trying to get calories consumed below calories needed.
> Dieting is calorie counting in some way or the other.
Atkins, and many similar low-carb diets, are not.
> It’s trying to get calories consumed below calories needed.
Even if you restrict “dieting” to “dieting for weight loss”, where that sentence is true, it's still not always some form of calorie counting. The goal of achieving a particular calorie balance is not always met through counting calories, even in rough proxy form.
Dieting for weight loss can have lots of strategies.
Atkins and other low-card diets appear to try to use the increase in energy expenditure (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5568065, Figure 2, they do a meta analysis but conclude "the effect sizes are so small as to be physiologically meaningless" so I'm not sure if the effect is real) due to the absence of carbs, and then just hope the person doesn't manage to increase the calorie consumption by a similar amount by eating too much---so it might still be called calorie counting, implicitly. I think you're defining calorie counting as specifically when the person following the diet counts the calories explicitly themselves.
You need to get your calories below your BMR in order to lose weight. Even on Atkins or any low carb diet, if you're consuming 3000 calories a day, you will not lose weight. Simple calorie counting could be deciding not eat a 36oz steak for dinner because of the high calories or skipping the 3rd piece of bacon for breakfast.