Part of the tricky thing about this circle of ideas is that several definitions 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 definition 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" in my post), 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" in my post; 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.
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.