  The depinning transition of elastic interfaces with an elastic interaction
kernel decaying as $1/r^{d+\sigma}$ is characterized by critical exponents
which continuously vary with $\sigma$. These exponents are expected to be
unique and universal, except in the fully coupled ($-d<\sigma\le 0$) limit,
where they depend on the "smooth" or "cuspy" nature of the microscopic pinning
potential. By accurately comparing the depinning transition for cuspy and
smooth potentials in a specially devised depinning model, we explain such
peculiar limit in terms of the vanishing of the critical region for smooth
potentials, as we decrease $\sigma$ from the short-range ($\sigma \geq 2$) to
the fully coupled case. Our results have practical implications for the
determination of critical depinning exponents and identification of depinning
universality classes in concrete experimental depinning systems with non-local
elasticity, such as contact lines of liquids and fractures.
