Definition: Statement to the effect that in certain conditions C a term A may be replaced by a term B and conversely.
This seems the clearest basic definition of "definition". Two normal reasons to justify this are that, in those conditions C, the meaning of A and the meaning of B are the same, or A is just a conventional abbreviation for B, with the same import. The reason this is then a formally valid inference is that terms with the same meaning have the same denotations, and hence statements P and Q that are all the same except that P has at one or more place a definition for A instead of A must have the same truth-value in the same conditions.
It should also be clear that the phrase "in certain conditions C" covers a lot, and not too clearly. One reason for this fact is that as soon as we have propositional attitudes, one person may know that a certain term A is defined by B and another may not, and so the substituting of defining for defined terms does not unproblematically produce statements of the same truth-values for all persons.
In any case it makes sense to regard definitions as explicit assumptions in arguments until one has proved that the definition can be proved as an equivalence.
1. Descriptive vs. stipulative definitions
There are two basic kinds of definitions: descriptive definitions, that attempt to describe in what sense a certain term is used in a certain society, and stipulative definitions, that propose a sense for a certain term, possibly regardless of the sense(s) the term has in any given society.
Quite often it is not clear for a given definition to what extent it is descriptive or stipulative, in as much as even most honest rendering of the various usages and meanings of a term in a society will contain some stipulative elements, and in as much even clearly stipulative definitions will normally follow some of the received meanings of a term.
In this Philosophical Dictionary it makes by far the most sense to consider each and every definition a stipulative definition, even if it is quite close to some received usage or definition of the same term.