Among models of $\lambda$-calculus, some like the Bohm tree model have the property that every element is a directed sup of definable elements, whereas others like the $D_\infty$ and $P(\omega)$ models contain "extra" elements (e.g. step functions) that are not sups of sets of $\lambda$-definable elements.

Does this property have a common name? A model is ___ if every element is a sup of a set of definable elements.

Related properties: separability, full-abstraction, completeness, standard.