There appear to be two different nested interval properties for the reals with the punchline "... then the intersection of the intervals is non-empty", and I'd like to know their respective histories (and the appropriate terminology to use for distinguishing between them). Here I am not concerned with the issue of "singleton-icity" of the intersection; only non-emptiness.
In one version (which I believe is due to Cantor), the length of the intervals is assumed to shrink down to 0; in the other version, their lengths are not constrained.
One sense in which the two properties are inequivalent is that there are non-archimedean ordered fields that satisfy the first but not the second. (Side question: Which is the preferred spelling, "archimedean" or "archimedian"? The number of Google hits for each spelling is about 800,000, so we can't use the Google-hits heuristic to decide, though Google seems to think that the former is preferred.) Specifically, consider the ordered field of formal Laurent series in $x$ over the reals, "ordered by size at zero" (so that effectively we're adjoining a formal infinitesimal $x$ to the reals). This ordered field satisfies the first nested interval property but not the second. (Consider the closed intervals $[nx,1/n]$. Their intersection is empty, but this doesn't contradict the first version of the nested interval property because the lengths of these intervals don't shrink to 0; indeed all the lengths are greater than $x$.)