I don't think that's quite right. There are two possible precise statements for such claims: 1) There exists some n such that all integers >= n satisfy the des

I don't think that's quite right. There are two possible precise statements for such claims: 1) There exists some n such that all integers >= n satisfy the desired property. 2) For n = [a specific constant], all integers >= n satisfy the desired property. I'm not familiar with Artin's conjecture, but from your description, it satisfies (1) but not (2). The reason is that if there are at most 2 such primes, we can take n to be the larger of the two primes plus one. Since all primes are finite, this choice of n is also finite. I think the key question is whether the paper described in this article proves (1) or (2). From the discussion, it sounds like it proves (2), which is the stronger result.