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.