The OP didn't invent the definition of fields as defined by additional and multiplication, with division defined as the Inverse of multiplication. Even the Wikipedia article on fields (https://en.wikipedia.org/wiki/Field_%28mathematics%29#Defini...) defines them this way. (Not that Wikipedia is an authority on truth, but I find that it's a pretty good indicator of what positions are common/widespread regarding a subject.) It's a bit silly to claim that division is defined as axiomatic "in mathematics" given how fundamental fields, defined as the OP defined them, are to modern number theory and mathematics in general.
In your link, division is defined as the inverse of multiplication. However, the inverse of multiplication is one of the axioms of the field, so by transitivity, division is an axiom as well (it's simply a convenient label for "the inverse of multiplication"). One can put the name "Division" to some other axiom or derived property, absolutely, but then "the inverse of multiplication" still needs to be dealt with. The author, if you note, conveniently omits the multiplicative inverse from their stated axioms.
I'm far from a pure mathematician, but even I can see through the facade, implying that it's pretty thin.
