On the one hand it is suggested that calories are calories, and that
low carb diets work simply because they result in less calorie intake.

Note the difference in terminology. "Calories are calories" is
actually used to mean "intake calores are intake calories."

It is commonly also alleged, though not clearly stated, that output
calories are NOT all equal. For example, it is often stated that
dieting without exercise will result in muscle loss while dieting with
exercise will result in fat loss, which is more desirable.

I am not sure that is proven either. But if true, then clearly output
calories are not all equal. Output through work is not equal to
output through excretion, e.g. And to the extent that input calorie
choice may affect output calories, the equality of input calories
begins to be questionable.