Clarification needed in Definability versus Leibnizian structure chapter on Hamkins's book

Question

In Lectures on the Philosophy of Mathematics, chapter 1, 1.10 Structuralism, there is the sentence:

For example, the real ordered field $<\mathbb{R},+,·,<,0,1>$ is Leibnizian, since for any two distinct real numbers...

The question I have is what it really means "the real ordered field $<\mathbb{R},+,·,<,0,1>$" in this context?

Or in other words, which tools I have at my disposal to "prove" it is Leibnizian?

I guess at minimum it means there is a first order language with equality and logical symbols, and signature $"<+,·,<,0,1"$, with enough variable symbols that allows me to write first order formulas.

I suppose I have intrinsically the first order non-logical axioms that "defines" the concepts of ordered field, and in consequence, I have the first order deductive aparatus to use the non-logical axioms to prove things in the object theory.

But I think I have more than that, isn't it? The original statement "the real ordered field..." implies that I have a concrete model in some background theory, isn't it?

Like I have a concrete structure defined maybe using ZFC or Second Order Arithmetic that somehow defines what this structure is in its own terms (in the background theory terms).

It doesn't really matter which background theory we are using to define "the real ordered field" just because it is categorical?

An last, loosely speaking, it seems as if the first order object theory is the weakest model, in the sense it can only prove whatever it is true in any model. In the other side a concrete model in the background theory, in general can "prove" things that the object theory can't. If for example another model can "prove" the negation of this thing in its background theory, then the thing becomes undecidable in the first order theory.

Answer 1

The next quote from Wikipedia explain my issue:

Structures are sometimes referred to as "first-order structures". This is misleading, as nothing in their definition ties them to any specific logic, and in fact they are suitable as semantic objects both for very restricted fragments of first-order logic such as that used in universal algebra, and for second-order logic.

In short, every time I read structure I automatically assumed it was a "first-order structure".

Nevermind, with Izaak comment and this, now it is easier to follow the arguments... :)