I will address the the most recent version of the question, which asks about the relationship between the following two features of a logic $L$, but note that a careful discussion of this topic should first clearly define what counts as a logic.

**(1) Abstract Completeness of** $L$: The set of valid $L$-sentences is recursively/computably enumerable [hereafter: r.e.].

**(2) Compactness of** $L$: A set $S$ of sentences of $L$ has a model if every finite subset of $S$ has a model.

**(1)** does not imply **(2)**.

For example, let $Q$ be the quantifier expressing "there are uncountably many", i.e., $Qx\phi(x)$ holds in a structure $\cal{M}$ with universe $M$ iff the set of $m\in M$ such that $\phi(m)$ holds in $\cal{M}$ is uncountable. Let $L_{FO}(Q)$ be the result of augmenting first order logic $L_{FO}$ with the new quantifier $Q$. Vaught proved that the set of valid sentences of $L_{FO}(Q)$ is r.e. Later [1970] in a seminal paper, Keisler gave an elegant axiomatization of $L_{FO}(Q)$.

On the other hand, it is easy to see that $L_{FO}(Q)$ does not satisfy compactness, e.g., for $\alpha < \aleph_1$ introduce constant symbols $c_{\alpha}$ and consider the set $S$ of sentences consisting of $\lnot Qx (x=x)$ [expressing "the universe is not uncountable"] plus sentences of the form $c_{\alpha}\neq c_\beta$ for $\alpha < \beta < \aleph_1$. It is easy to see that every subset of $S$ has a model, but S itself does not have a model.

I should point out that $L_{FO}(Q)$ has a limited form of compactness known as *countable compactness*: if $S$ is a *countable* set of sentences of $L_{FO}(Q)$, then S has a model if every finite subset of $S$ has a model [Vaught, ibid].

All of the above features of $L_{FO}(Q)$ [abstractly complete, countably but not fully compact] are shared by a number of other generalized quantifiers, including the *stationary quantifier* [introduced in the late 1970's and intensely studied in the 1980's]. However, as shown by Shelah, there are other generalized quantifiers that generate fully compact logics that also are abstractly complete

**(2)** does not imply **(1)** either.

For example consider the "logic" whose nonlogical symbols are the arithmetical ones, and whose axioms are the usual axioms of first order logic plus all the axioms of *true arithmetic* , i.e, arithmetical sentences that hold in $\Bbb{N}$. The semantics of the logic is the same as first order logic, so compactness continues to hold; but clearly abstract completeness fails by Tarski's undefinability of truth-theorem, which says that $Th(\Bbb{N})$ is not arithmetical, let alone r.e.

**PS** A "naturally occurring logic" that also serves to show that **(2)** does not imply **(1)** is the *existential fragment of second order logic*; its compactness follows from the usual proofs of compactness of first order logic [including the ultraproduct proof], but its set of valid sentences is known to be co-r.e., but not r.e.