Definition 1.2 in <A HREF="https://arxiv.org/abs/1303.1421">On the Distributional Hessian of the Distance Function</A> explains the difference: an inequality can hold in distributional sense, in barrier sense, or in viscosity sense.

It is stated that the definition of an inequality “in barrier sense” goes back to Calabi (1958, he used the terminology “weak sense” rather than “barrier sense”).