Formally write a definition using quantifiers and predicate logic

Question

I quote a passage from Ted Sundstrom's Mathematical Reasoning: Writing and Proof:

A note about Definitions: Technically, a definition in Mathematics should almost always be written using "if and only if". It is not clear why, but the convention in mathematics is to replace the phrase "if and only if" with an "if" or an equivalent. Perhaps this is a bit of laziness or the "if and only if" phrase can be a bit cumbersome. In this text we will often use the phrase "provided that" instead

The author goes on to provide an example of a definition:

A nonzero integer $m$ divides an integer n provided that $(\exists q \in \mathbb Z)(n=m\cdot q)$

I have a couple of questions:

I don't really buy that the use of an equivalent to substitute "if and only if" is due to "a bit of laziness". Which one should it be for a formal (i.e. syntactically well defined/formal) definition?
How would I write the definition above symbolically using the appropriate syntax?

Basically, a definition is an "iff" statement; it rules the introduction of a new "term" (the definiendum) that is "equated" to a previously existent term (the definiens). In some contexts this "equation" can be symbolized with $=_{\text {def}}$ — Mauro ALLEGRANZA, Apr 08 '21 at 10:23
Thanks... I am now trying to make sense of the other links you posted. For your proposed definition, why are you not providing quantifiers for $m$ and $n$? — user32882, Apr 08 '21 at 11:43
It seems to me that a definition is neither an open sentence (there is no "truth set" to a definition) or a statement (a definition does not evaluate to true or false). In that case what is it? And why do quantifiers appear in it? I was under the impression that the entire role of quantifiers is to turn open sentences/predicates into statements... — user32882, Apr 08 '21 at 12:01
I would argue that makes the statement false, since not every integer $n$ is divisible by every integer $m$ — user32882, Apr 08 '21 at 12:15
Yes, therefore $\forall m \forall n ( \text {Div}(m,n) \leftrightarrow \ldots)$ is false — user32882, Apr 08 '21 at 15:25

Formally write a definition using quantifiers and predicate logic

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