Thanks for the references and the examples Stefan! I have already look them up. Coming back to example no 3 above: I certainly want to have a setup that includes the case of smooth functions on a manifold. Can one find a nice class of rings / algebras (including algebras of smooth functions on a manifold) for which "theorems about positive matrices" hold? What is the crucial property that one needs? Maybe it is technically easier to stick to a positivity defined through positive functionals?

Thank you for the remark! Unfortunately, I'd like to answer "no" to the first two questions. I'd like the ordering to be a rather weak one, similar to the "positive cone" of $\ast$-algebras, i.e. elements that can be written as sums of $a^\ast a$ (or just sums of squares with the trivial star involution). This will induce a "ring pre-order". Furthermore, I don't want to restrict myself to integral domains; I have many examples with zero divisors, that I'd like to consider.