Is there a version of the Archimedean property which does not presuppose the Naturals?

It is not surprising that some versions of the Archimedean property concern subsets of the order rather than merely elements. The reason is that the Archimedean property is provably not expressible in a first order manner.

This is because the structure of the reals R, as an ordered field, say, (but one can add any structure at all), has elementary extensions to nonstandard models R^* which are non-Archimedean. This means that any statement in the language of ordered fields that is true in the reals R will also be true in the nonstandard reals. To prove that such models exist is an elementary application of the Compactness theorem, and one can also construct them directly via the ultrapower construction. One can also control the cofinality of the nonstandard order. For example, one can arrange that every countable subset of R^* is bounded. Since all these various nonstandard models R^* satisfy exactly the same first order truths as the standard reals R, but are non-Archimedean, it follows that being Archimedean is not first-order expressible.

Being Archimedean is, of course, second-order expressible, and the usual definition is a second order definition. As Neel mentions in the comments, the natural numbers are identifiable as the smallest subset of the ordered field containing 0 and closed under successor n+1.

If one adds the natural numbers N as a predicate to the original model, so that one is looking at R as an ordered field with a unary predicate holding of the natural numbers, then the nonstandard model R^* will include a nonstandard version N^* of the natural numbers. This new field R^*, which is not Archimedean, will nevertheless appear to be Archimedean relative to the nonstandard natural numbers N^*. For example, for any x and y in R^*, there will be a number n in N^* such that nx > y.

Indeed, one can do amazing things along this line. Suppose that V is the entire set-theoretic universe, and let V^* be a nonstandard version of it (such as an ultrapower by a nonprincipal ultrafilter on the natural numbers). Inside V^*, the structure R^* is thought to be the actual real numbers and so V^* thinks R^* is Archimedean, even though back in V we can see that it is mistaken, precisely because V^* is using the wrong set of natural numbers for its conclusion. The model V^* simply cannot see the true set of natural numbers sitting inside R^*, because it does not have that set.

More generally, one can similarly describe what it should mean for any ordered field F to be Archimedean relative to a subring R. Perhaps this simple idea is the generalization for which you are looking? It is mainly amounting to the question of whether the subring is cofinal in the original order.

Thus, it is very natural to look at the possible cofinalities of the orders that arise in ordered fields (or the other types of structures that you consider). For any infinite regular cardinal κ, one may find an elementary extension of the reals R to a nonstandard ordered field R^*, where the order of R^* has a cofinal κ sequence. To do this, just perform a series of κ many extensions, each with new elements on top of the previous model. In κ many steps, the union of the structures you built will have an order with cofinality exactly κ.

If one only uses the ultrapower construction to construct the nonstandard models, however, then there are limits on the resulting cofinality of the order. Understanding these limits is a large part of Shelah's deep work on PCF (= possible cofinality) theory.

Tarski showed the first-order theory of the reals is decidable. But of course the first-order theory of the naturals isn't. So (assuming consistency, I guess) we know you cannot construct the naturals in the reals using only first-order notions. But we can construct the natural numbers if we are allowed to quantify on sets of reals.