I think the notion of "positive" in this question indicates that a constructive definition is requested, as opposed to an existence proof. The unsatisfying part of the traditional definition is that given some particular value $x$, I cannot easily tell whether it's irrational or not. Furthermore, if I want to start enumerating the irrationals, the definition helps me not at all.

For this reason, I think that Eric Wofsey's answer is the only one which satisfies the spirit of the question. It *both* constructs instances of irrational numbers *and* offers a method for enumerating them!

From a computational perspective, all definitions require an infinite amount of work to verify the irrationality of $x$, due to the nature of irrational numbers to begin with. But with the continued fraction definition, one can compute increasingly longer approximations which get closer and closer to $x$, which are the convergents of $x$. With the other definitions, it is not clear how to proceed in a manner which increases one's confidence in the irrationality of $x$.

That is, if $x$ is actually rational, then there must be some $n$ at which the sequence equals $x$, and it should always be clear how to adjust the sequence to get closer to $x$. That's because increasing $a_i$ will always produce a bigger change than increasing $a_j$ when $i < j$. Therefore, searching for the sequence $a_0...a_n$ which is closest to $x$ should be linear in $n$.

Of course, this is hand-waving a bit, since this process requires a computer which can represent arbitrary reals to begin with, even though there can be no such thing. But the argument still works if we bound the representation by saying that we can only inspect the first $B$ digits of each value.

More importantly, for rational values of $x$, we know that computing the best rational approximation *will* terminate in finite time, and that this time is proportional to the "irrationality" of $x$, where I define "irrationality" to be the number of values $n$ in the continued fraction sequence which exactly equals $x$ (meaning, all successive values are 0). When $n = \infty$, $x$ is "completely irrational" or just "irrational". Of course, when $n = 0$, then $x$ is "completely rational", or "integer". When $n$ is small, $x$ is only "slightly irrational". ;)

On the other hand, it is not at all apparent to me how one could use Dirichlet's or Hurwitz's theorems to determine the rationality of $x$ in finite time, let alone an algorithm which is linear in the irrationality of $x$.