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 or an identity. 

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.

2. Creative vs. uncreative definitions

The distinction in the previous section is related to the distinction between creative vs. uncreative definitions. Formally speaking, which means here especially: with reference to some presupposed formal system of reasoning, an uncreative definition in such a system corresponds to an equivalence that is instantiated: There is - according to the assumptions of the system - something that is defined in terms of the equivalence, and that is consistent with the rest of the system.

Accordingly, an uncreative definition corresponds to a piece of terminology that circumscribes what the term stands for, but that can be wholly removed or avoided in as much as the defined term can be replaced by its equivalent defining expression.

By contrast, creative definitions embody some assumption, usually to the effect that there is something that corresponds to the defined term. Without a fully formalized system it is often not easy to prove or see whether a definition used in the system is creative or not. And the reason for the term "creative" is that a creative definitions entails statements that are provable in the system, but that are not provable if the definition is removed, which is to say, in other words, that the used equivalence in the definition is not provable in the system.

A proper axiomatic system is generally supposed to be free from creative definitions, since a proper axiomatic system lists all its assumptions explicitly as axioms (and doesn't - consciously or not - smuggle them inside by what seem to be mere terminological conventions).