When to write something in terms of axioms? Most mathematical structures are defined according to axioms. e.g. we state:

Definition. Monoid. A monoid is a tuple $(S,\cdot)$ where $\cdot$ is a binary operation $S\times S\to S$ that satisfies the following axioms:

*

*Axiom 1. Associativity. For all $a, b, c \in S$, it holds that $(a\cdot b)\cdot c=a\cdot (b\cdot c)$


*Axiom 2. Identity element. There is an element $e\in S$ such that for every $a\in S$, we have $a\cdot e=e\cdot a =a$.

But we could also have written this in the following setup, without using "axioms".

Definition. Associative operation. A binary operation $A\times A\to A$ on some set $A$ is called "associative" if it holds for all $a, b, c \in A$, it holds that $(a\cdot b)\cdot c=a\cdot (b\cdot c)$
Definition. Identity element of an operation. A binary operation $A\times A\to A$ on some set $A$ has an "identity element", if there is an element $e\in S$ such that for every $a\in S$, we have $a\cdot e=e\cdot a =a$.
Definition. Monoid. A monoid is a tuple $(S,\cdot)$ where $\cdot$ is a binary operation $S\times S\to S$ that is associative and has an identity element.

It is my understanding that there is only an aesthetic difference between these two. That is, we could effectively remove the whole concept of "axiom", and replace all texts in all of mathematics that are written in the first style with texts written in the second style. Mathematics would not change content-wise if we did this.

However, my question is not about generally known concepts like monoids, but about concepts that the author themselves have come up with.
My question is essentially: When someone writes a paper, and comes up with a new concept, when should he/she write it in the first style, and when in the second style? i.e. Are there norms for when authors should call their new concept an "axiom", rather than simply giving it a name and a definition?
I can imagine that it is frowned upon to write "axiom" for something that is not a very fundamental concept, even though calling it such has no implications whatsoever content-wise. E.g. let's say I come up with a very specific type of group which satisfies some very obscure property, would it then be considered "arrogant" to call that property an "axiom", instead of just saying "An [some adjective] group is a group such that ..."
Are there generally accepted rules for when to call a new concept an "axiom", and when to write it in the second style?
 A: There are not any strong norms of this sort.  It's definitely not "frowned upon" or considered "arrogant" to call something unimportant an "axiom".  To the extent that there are some conditions where it is more common to use "axiom" than others, it just a matter of habit and the vague connotations of "axiom".
Some situations where it is particularly common to use the term "axiom" include:

*

*When you are defining a general type of mathematical structure of broad importance, at least in the context of your work.  So, for instance, it would be much more common to use the term "axiom" in the main definition of what your entire paper is studying than it would be in the definition of some technical tool you are using in a small lemma.

*When you actually are "axiomatizing" something, in the spirit of the original historical meaning of "axioms".  That is, you are providing some abstract assumptions which can be taken as a foundation for some broader theory, which is often "categorical" (i.e., unique up to isomorphism) or nearly so.  A relatively modern example of "axioms" in this sense would be the Eilenberg-Steenrod axioms for (co)homology, for instance.

*When you are studying the structures you are talking about from a point of view related to logic (e.g., in model theory or universal algebra, it is common to refer to "axioms" for some kind of first-order structure).

*When you want to have a nice way to refer back to the individual axioms later, as discussed in Ted's answer.

I would add that whether or not you use the word "axiom", the style of your first definition is generally preferred, since it is more succinct and packages everything nicely in one place.  This is especially the case if the individual "axioms" are not particularly of interest on their own, and you only really care about the full definition.  For instance, here is how I might define monoids without using the word "axiom":

Definition. A monoid is a pair $(S,\cdot)$ where $\cdot$ is a binary operation $S\times S\to S$ satisfying the following two properties.

*

*For all $a, b, c \in S$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.


*There is an element $e\in S$ such that for every $a\in S$, $a\cdot e=e\cdot a =a$.

This is basically the same as your first definition, just with some wording tweaks and "axioms" replaced by "properties".
A: There is no difference whether you call them "axioms" or not, as long as you mention what properties you are assuming.  It is traditional to call them "axioms" because when you are studying (say) monoids, those are the assumptions you are working from.
(However, I should mention that the second style, as written, is grammatically weird because it is not normal for "where" to be followed by a new sentence. If I were writing in your second style, I would put the definitions of associativity and identity before the definition of monoid, and then end the definition of "monoid" before the word "where".)
Additional comment based on edit:
Calling something an axiom is not an indication of arrogance.  If you are defining a new structure and it satisfies some axiom which you name foo, you could say "foo axiom" or "foo property" or "foo law".  Or, you could just use "foo" as an adjective directly in front of some existing term (like "group"), as in one of your examples.  These are stylistic choices.  Giving something a name (like "foo") is already an indication that you think it's important in the context of whatever you are writing; adding a word like "axiom" doesn't change that. 
Using a term like "foo axiom", as opposed to just an adjective "foo", can be convenient if you need to refer to the axiom later independently of the definition it first occurs in.  For example:


*

*When you use the foo axiom in a proof later and you want to emphasize that the foo axiom is important for a particular step, you could write "By the foo axiom, ..."

*If you have several new axioms and want to study what happens when one of them is removed, you could say "if we remove the foo axiom, then ..."

*If you want to define several different things that all use the foo axiom, you can define "foo axiom" once and then just say "foo axiom" in all the definitions.
A: Whether or not you choose to use the word "axiom" I would strongly recommend the first version over the second. If you're writing about monoids it's very likely that your reader knows about associativity and unit elements. That makes the second version much too wordy. In most cases you would not even have to remind your reader about the meanings, so could say simply

Definition.  A monoid is a tuple $(S,⋅)$ where ⋅ is an associative
  binary operation $S×S→S$ with an identity element.

or even (a little less formal, but easy to understand)

Definition.  A monoid is set $S$ with an associative
  binary operation (which we write as $x \cdot y$) with an identity element.

The  moral: when writing (mathematics or anything else) think about what would be easiest for your intended readers. Sometimes there are generally accepted rules that help with that, but not always.
Edit in response to an edit in the question. 
I would give the same advice even if you were defining a structure no one had ever seen before and decided to call it a monoid.
