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@Mr.Wizard Grid should work just as well.

Sigh... RandomPoint for intersections of hyperplanes and cuboids of dimension larger than 3 stay unevaluated...

And for implicit region specification of the same, three dimensions is already too much.

Eh. Two different versions of Mma with two different licenses on my laptop decided to use each others' licenses and one started complaining about coming license expiration. Is this common behaviour (on Mac)?

@kirma Never happened to me (I had two licenses at one point.)

Also, it went away as mysteriously as it appeared. I wonder if there's a license pool somewhere or something and I had some instance that I didn't even remember on the background, etc.

@CATrevillian Yet another way to achieve this (is a rather hacky/ugly manner) is what I did in MergeRules in my paclet - essenitally it does something like

(match1:pat1)|(match2:pat2):>With[
  {matches={{match1},{match2}}},
    matches[[1,First@FirstPosition[matches,{__}]]]
  ]

i.e it labels the individual alternatives, and gives you the first non-empty match

looking at it again, you might be able to do something cleaner like this:

(match1:pat1)|(match2:pat2):>With[
  {matches={{match1},{match2}}},
    FirstCase[matches,{match__}:>match]
  ]