They're not equivalent. For example, Lurie-compact objects in a category of $R$-modules are the same as finitely presentable modules. (The same is true for any category of algebras for a Lawvere theory, i.e., an algebraic theory whose operations are finitary, subject to universally quantified equational axioms.) On the other hand, Murfet-compact objects in a category of $R$-modules need not be even finitely generated (although they will be if $R$ is Noetherian). There was a fairly long discussion about this here: "Sums-compact" objects = f.g. objects in categories of modules?

Different communities use different terms. The term 'compact' is in some ways suggestive, but I don't think it's optimized.

They're not equivalent. For example, Lurie-compact objects in a category of $R$-modules are the same as finitely presentable modules. (The same is true for any category of algebras for a Lawvere theory, i.e., an algebraic theory whose operations are finitary, subject to universally quantified equational axioms.) On the other hand, Murfet-compact objects in a category of $R$-modules need not be even finitely generated (although they will be if $R$ is Noetherian). There was a fairly long discussion about this here: "Sums-compact" objects = f.g. objects in categories of modules?

Different communities sometimes use the same term differently. The term 'compact' is in some ways suggestive, but I don't think it's optimized.