Part of the tricky thing about this circle of ideas is that several definitions are not equivalent in full generality but become equivalent with extra hypotheses. For example, a basic result about compact objects is the following characterization of module categories, which among other things provides a characterization of Morita equivalences.

**Theorem (Gabriel):** A cocomplete abelian category $C$ is equivalent to the category $\text{Mod}(R)$ of modules over a ring $R$ iff it admits a compact projective generator $P$ such that $\text{End}(P) \cong R$.

Both "compact" and "generator" in the statement of this theorem are individually ambiguous. "Compact" could mean either Lurie-compact or Murfet-compact, and "generator" can have something like ~7 different meanings, maybe ~3 of which are in common-ish use (?); see Mike Shulman's *Generators and colimit closures* (which discusses 5 possible definitions) and my blog post *Generators* (which discusses 6 possible definitions, 4 of which overlap with Mike's) for a discussion.

The happy fact is that nevertheless, the meaning of "compact projective" and of "compact projective generator" in the statement of Gabriel's theorem is unambiguous:

- in a cocomplete abelian category, "compact projective," using either Lurie-compactness or Murfet-compactness, is equivalent to the condition that $\text{Hom}(P, -) : C \to \text{Ab}$ commutes with all (small) colimits (this condition is also known as being tiny; see my blog post
*Tiny objects* for a discussion), and
- for compact projective objects in a cocomplete abelian category, nearly all of the definitions of "generator" that I'm aware of collapse and become equivalent. I'll limit myself to naming two: the weakest is that every nonzero object admits a nonzero map from $P$ (which I call "weak generator"; I forget if this name is standard), and the strongest is that every object can be written as the coequalizer of a pair of maps between coproducts of copies of $P$ (which I call "presenting generator"; this is not standard. In an abelian category coequalizers can be replaced with cokernels but this definition generalizes nicely to algebraic categories such as groups and rings).

There is the additional nuance that in a stable $\infty$-categorical setting like the one Lurie works in it seems that one can drop projectivity but I'm not sure what the precise statements are. E.g. I believe there's a stable $\infty$-categorical analogue of Gabriel's theorem characterizing module categories over $E_1$ ring spectra and I believe that analogue involves compact generators.

Anyway, for what it's worth I would advocate for Lurie-compactness as the "default" meaning of compactness. Murfet-compactness is quite specific to the abelian setting, but Lurie-compactness is nice in many settings; for example, in the category of models of a Lawvere theory (groups, rings, etc.) an object is Lurie-compact iff it's finitely presented. Already this implies the not-entirely-obvoius fact that for modules being finitely presented is Morita invariant.