You're interpreting the term too loosely in the second part of your question. Your first definition "the connection between words and meaning of those words" is acceptable for logic if you add "described in a precise, rule-governed way". Logicians speak of terms, formulas, operators, connectives, and quantifiers rather than mere words, though.
Unfortunately, when you step away from such informal definitions and take a closer look things become less clear. I've seen the term being used in at least three slightly different senses. The first one is the general case, whereas (2) and (3) are imprecise. However, when someone talks about "the semantics" of a particular formal system, he may mean any of (1)-(3):
(1) In a first, primordial sense, "semantics" is used for "the definition of possible interpretations" and "the definition of models for a formal language". The semantics of a logical system is a set of rules for interpreting well-defined expressions of a formal language, which is in turn defined by syntax rules (a grammar). A logical semantics usually leaves it open how extralogical expressions are interpreted and focuses on a particular subset of expressions of a language called logical expressions. These are for example terms (how to deal with them in general), operators, connectives, quantifiers, and formulas. For the extralogical expressions such as particular interpretations of terms, you may then provide different models within the range circumscribed by the general semantic rules (aka interpretation, evaluation rules).
(2) In a second, narrower sense based on (1), "semantics" sometimes abbreviates denotational semantics which is also sometimes called model-theoretic semantics. This involves the explicit mapping of expressions of the language to elements in sets and rules for counting and combining these elements expressed in the language of set theory or, in the simplest case, by providing truth tables. So here "semantics" is understood as (1)+a particular method of achieving (1).
(3) In a third sense closely related to (1), "semantics" is meant in various ways that may be precisely described as algebraic semantics, combinatorial semantics, category-theoretic semantics, and so forth, where the methods used aren't based on set theory and are generally more syntactic in nature. ("syntactic" is meant as symbol manipulation by means of rules here.) Again, this sense of "semantics" is (1)+ = a particular method of achieving (1). These kinds of semantics are not very far from proof-theoretic methods such as using axiom systems + deduction rules, natural deduction, tableaux which are commonly considered syntactic methods. As confusing as this might be, speaking of "semantics" despite a certain nearness to proof theory may be well-justified in such cases provided that the method is provably more powerful than any calculus could be.
The third use of "semantics" often conveys an overall proof-theoretic attitude of a logician, the view that the main business of logic (as opposed to mathematics) is to provide the meaning of expressions of logical languages by a well-behaved proof theory only. Logicians in this tradition will sometimes even speak of proof-theoretical semantics.
Things get really messy when you go beyond first-order logic to second-order logic with standard models, because second- and higher-order logics with standard models have no complete proof theory (axioms+symbol manipulation rules), and in that case which kind of semantics you choose can make a huge difference both conceptually and formally.
