"Zero" is just a specific element of the field which satisfies being the additive identity as well as the rest of the properties of a field where 0 is mentioned

"Zero" is just a specific element of the field which satisfies being the additive identity as well as the rest of the properties of a field where 0 is mentioned. When the rest of the axioms refer to "zero" they refer to the exact same element of the set that is also the additive identity. It's not the "number zero" because a field does not care about numbers, it's just elements of a set (which might be numbers like in R's case). 1 is not "multiplicative zero", it's the "multiplicative identity". 0 and 1 are just the shorthand we give for those elements. because those are the symbols we use in R which is the most common field we deal with in everyday life. Or am I misunderstanding your question?