In Theorem 5.19 of Kelly's Basic Concepts of Enriched Category Theory, it is proven that a fully faithful functor $K \colon \mathcal A \to \mathcal C$ is dense if and only if $\mathcal C$ is the closure of $\mathcal A$ under colimits of a family of diagrams, by which is meant a class of pairs $(F \colon \mathcal L^\circ \to \mathcal V, P \colon \mathcal L \to \mathcal C)$ consisting of a weight and a diagram for that weight.

Typically, cocompletions of (enriched) categories are taken with respect to a class of weights, rather than a family of diagrams. However, I have found it difficult to find explicit comparisons between the expressivity of these approaches. I have several related questions.

1. Is it possible to characterise those fully faithful functors $K \colon \mathcal A \to \mathcal C$ that exhibit $\mathcal C$ as a closure of $\mathcal A$ under $\Phi$-weighted colimits (for some class of weights $\Phi$)? Is this concept strictly less general than density?

Assuming closure under weighted colimits is less general than closure under colimits of a family of diagrams:

2. What are some natural examples of dense functors that are only captured by considering diagrams rather than weights?
3. What are the difficulties in considering cocompletions with respect to families of diagrams, rather than cocompletions with respect to classes of weights?

I looked through several papers on these concepts, but couldn't find the answer in them, e.g. Albert–Kelly's The closure of a class of colimits; Kelly–Schmitt's Notes on enriched categories with colimits of some class; Kelly–Lack's On the monadicity of categories with chosen colimits.