Difference between a "theory" in logic and a "system of axioms"
Solution 1:
It may help clarify the issue with "axioms" by looking at how the meaning of that word has changed. In Euclid's Elements, and for a long time after that, an "axiom" or "postulate" was not just any sentence: an axiom had to be obviously true and self-evident, so that no proof was required. In this traditional sense, the negation of the parallel postulate would not qualify as an "axiom", because it's not obviously true. For example, the fuss over the parallel postulate started because it wasn't clear that the parallel postulate was sufficiently self-evident.
In modern logic, we worry much less about the "self-evident" requirement [1]. When we are working in complete generality, any set $S$ of sentences can be regarded as a set of axioms. The set of all sentences that can be deduced from $S$ is then the deductive closure of $S$. With this reductive meaning of "axiom", there is no longer much difference between a theory and a set of axioms. We could consider every sentence in the theory to be an axiom, for example, while Euclid would not accept every statement provable form his postulates as a postulate.
The word "theory", as matt says in his answer, is used in several ways. Sometimes it is used to mean a deductively closed set of sentences, and sometimes it is used to mean just any set of sentences, which might also be called a set of axioms. In most settings, we can replace a set of axioms with its (unique) deductive closure when necessary, so the difference between the two conventions is not very substantial.
There is one more meaning of "theory" I want to point out. I mentioned theories that are generated by taking the deductive closure of a set of sentences that are treated as axioms. Another way to form a theory is to take some class of semantic structures, in the same signature, and then form the set of all sentences that are true in all structures of the class. For example, one can form the "theory of abelian groups" and the "theory of the real line". Such theories will always be deductively closed. The difference here is that we did not start with a set of axioms.
[1] We do worry about self-evidence when we are trying to justify a foundational theory such as ZFC. And the axioms we assume are often obviously true, in which case there is no issue. But we also look at axioms like the axiom of determinacy, which is disprovable in ZFC. Some of the traditional usage remains, but only in certain contexts.
Solution 2:
The fact is that the meaning of the term theory varies within mathematical logic, and different logicians and logic texts define this term differently. Some say that a theory is any set of sentences, others insist that it be deductively closed. You asked for a bit of notation, and I have seen various ways of denoting the deductive closure of a set $T$ of sentences, such as $\text{Cons}(T)$ for "conseqeuences" and also $\text{Thm}(T)$ for "the theorems of $T$".
None of the principal concepts that we apply to theories depend on the difference, and logicians are usually happy to move from one definition to another (e.g. at a lecture) with ease.
A theory $T$ is complete if $T\vdash\varphi$ or $T\vdash\neg\varphi$ for every $\varphi$.
A theory $T$ is consistent if it does not derive a contradiction.
A theory $T$ is satisfiable if there is a model $M\models T$.
A theory $T$ is finitely axiomatizable if there is a finite set $T_0$ derivable from $T$ that also proves all of $T$. (One may assume without loss that $T_0\subset T$.)
A theory $T$ is computably axiomatizable if there is a computably decidable set of axioms $T_0$ derivable from $T$ that also proves all of $T$. (By an interesting observation of Craig, this is equivalent to the assertion that the set of theorems of $T$ is c.e., or that $T$ has a c.e. set of axioms.)
All of the above properties work for either concept of theory, either as a set of sentences or a deductively closed set of sentences. One subtle differences arises with the last point (computable axiomatizability), since if you have the set-of-sentences understanding of theory, then you may not necessarily assume that $T_0\subset T$.
Let me mention also another similar issue, namely, that in common usage, as opposed to formal definition, many logicians use the term"*theory" to mean "consistent theory".
(Finally, I would like to second User6312 remarks opposing the casual exclusion in the question of logic from mathematics. )
Solution 3:
The two definitions are floating around even within logic. Some authors just say any set of sentences is a theory; some require them to be closed under deduction / semantic consequence.
It typically doesn't matter, since, for example, if $T$ is a set of sentences and $T\models \phi$, then (exercise) for every sentence $\psi$, $T\models \psi$ if and only if $T\cup \{\phi\}\models \psi$.
The more important distinction regarding theories containing or not containing sentences is between complete and incomplete theories (though, depending on how you define theories, you might see complete theories defined in terms of deduction or containment).
Solution 4:
The relationship between logician and mathematician has precisely the same character as the relationship between functional analyst and mathematician: a special case. More precisely, a mathematician is a logician when (s)he is doing logic. The same person may write papers that have a different subject classification.
And a theory, in first-order logic anyway, is a deductively closed set of sentences. This hardly needs saying, since if it is mathematicians' notion of theory, then it is a mathematical (and therefore logical) consequence of the fact that a logician is a mathematician.
Added, evidence: We talk about a theory being finitely axiomatizable. Surely this cannot mean that a set of axioms is finitely axiomatizable. There are a number of examples of this general character.
Also, there is sometimes discussion about alternate axiomatizations, of, for example, the theory of groups. This supports the notion that by the theory of groups we mean the body of theorems.
When we discuss model completeness, or completeness, of certain theories, we do not necessarily have a specific set of axioms in mind, since axiomatizations differ.