The treatment on the nLab seems likely to be the state of the art here. There's no generally accepted definition, and to get one you might have to generalize to the stuff-structure-property perspective in which every functor can be seen as forgetful. This makes functors such as the projections $\mathcal C\times\mathcal D\to\mathcal C$ "forget stuff," namely, the $\mathcal D$ object of a pair.
If the idea of forgetting stuff feels unnatural, then maybe you want forgetful functors only to forget properties and structure-but this includes all faithful functors.
I think it's clear that every fully faithful functor $F:\mathcal D\to \mathcal C$ can reasonably be called forgetful, since it forgets the property of being in the subcategory $F(\mathcal D)$.
I don't see that there's such a natural description of what $F$ forgets when $F$ is merely faithful, but neither do I know of any way to specify when $\mathcal D$ is a category of $\mathcal C$-objects with structure, which might describe the most natural class of forgetful functors, better than the existence of a faithful functor $\mathcal D\to \mathcal C$. The comments under this question discuss the case of defining categories structured over sets, though not conclusively-but I'm at least convinced that there's no extant definition better than faithfulness.
So, it appears your options are "all functors," "faithful functors," or this.