The notion of model site is a model category model for the notion of (∞,1)-site.
An (∞,1)-category equipped with the structure of a site on its homotopy category serves as a domain for ∞-stacks. It may be modeled by a model category.
A model category with a site structure on its homotopy category is called a pseudo model site (def 4.1.1 in ToVeI).