Axiomatization of $\mathbb{Z}$ via well-ordering of positives.

Second-order quantification allows us to talk about properties of subsets of the ring, much like the completeness axiom of the real numbers (which is too a second-order statement).

We can adjoin the usual theory of ordered rings the following axiom:

$$\forall A(A\neq\varnothing\land\exists x\forall a(a\in A\rightarrow x<a)\rightarrow\exists y(y\in A\land\forall x(x\in A\rightarrow y\leq x)))$$

Saying that for non-empty every set $A$, if there is a lower bound for $A$ then $A$ has a minimal element.

We can also follow Zhen Lin's suggestion in the comments. Notice that $\mathbb Z$ is the unique free additive group which has only one generator. That is: $$\exists x(x\neq 0\land\forall A(x\in A\land\forall a\forall b(a\in A\land b\in A\rightarrow a+b\in A)\land\forall a(a\in A\rightarrow -a\in A)\rightarrow\forall y(y\in A)$$

This is a very complicated way of saying that there exists some $x$ which is non-zero and every $A$ in which $x$ is an element, and $A$ is closed under addition and negation imply that $A$ is everything.

In $\mathbb Z$ this is true because $x=1$. However this is not true for any other ordered ring.

Any ordered ring R whose positives P are well-ordered in R is isomorphic to $\mathbb Z$ as an ordered ring. The proof is easy. Hint: $ $ the natural image of $\,\mathbb Z\,$ in R is an order mononomorphism, so it remains to show it is onto. If not, R has a positive element $\rm\:w\not\in \mathbb Z.\:$ $\rm w$ is not infinite $\rm (w\! >\! n,\, \forall\, n\in\mathbb N)\,$ else $\rm\,w > w\!-\!1 > w\!-\!2,\ldots\,$ is an infinite descending chain in P, contra P well-ordered. Therefore $\rm\:w\:$ must lie between two naturals $\rm\:n < w < n\!+\!1,\:$ hence $\rm\ 0 < \epsilon < 1\:$ for $\rm\:\epsilon = w\!-\!n,\:$ therefore $\rm\: \epsilon > \epsilon^2 > \epsilon^3 > \ldots\,$ is an infinite descending chain in P, $ $ contra P is well-ordered. $ $ QED

You ask for another example of a discrete ordered ring. As here, order the ring $\rm\,\mathbb Z[x]\,$ of integer coef polynomials by: $\rm\:f > 0\:$ iff it has leading coefficient $> 0,\,$ i.e. iff $\rm\:f\:$ is positive at $+\infty,$ and $\rm\:f > g\:$ iff $\rm\,f\!-\!g > 0.\:$ Here, as above, $\rm\:x > x\!-\!1 > x\!-\!2 > \ldots\, $ so its positives are not well-ordered.