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?