In a definition, when we say that X is called by the name Y, if it satisfies certain properties, we usually mean that entities in the same universe of discourse and of the same kind as X are not called by the name Y if they do not satisfy those properties.
is_called(X, "Y") is not a proposition in the domain of discourse; it is in the layer of symbols that are being used to discuss the domain, the "meta-domain", if you will.
Something in the actual domain cannot be put in a logical relationship with
something in the meta-domain; that is a level violation.
It is never sensible to say:
If a number is divisible by four, it is called "even".
because a definition is expected to be complete, and some numbers that are not divisible by four are also even. If we drop the "called", it's something completely different:
If a number is divisible by four, it is even.
Now the symbol "even" is not being quoted for the purposes of being defined; it is evaluated and replaced by the properties that it denotes, and so we have a logical relationship being expressed purely in the domain of discourse.
So, in a way, the "if" in a definition is related to "if and only if": clearly, a number is not called even if it is not divisible by two, therefore a number is called even if, and only if it is divisible by two. However, it is redundant and verbose. Moreover, definitions which use "if" can always be rewritten into ones which do not use if at all:
Integers divisible by two are called "even".
We can, and should, use "if and only if", whenever we give a definition in such a way that we are not quoting the name. Suppose I have never defined what it means to be "even". It is appropriate to say:
Integers can have certain property: they can be called even under certain conditions.
An integer is even if and only if it is divisible by two.
Here, the iff is necessary, because the second statement isn't asserting itself as a definition of the term. I didn't use the word "called" or similar.
One or more such statements can constitute an implicit definition of the term, when they specify enough rules that every integer can be classified as even or not even.
The use of iff in the second statement above informs the reader, "I am the only rule you need to deduce what set of properties are denoted by the term 'even'".