I love Dijkstra's writing, but I don't think this is his strongest piece. In general parlance, when we say "by piegonhole" we mean "any variant of it". I'd still call what he's doing "piegonhole" lol. You can even further generalize it, e.g. by making expected value arguments.

This is not uncommon: we can say that "by the fundamental theorem of algebra" two polynomials of degree N that agree on N+1 points are identically equal. "By induction" includes Cauchy induction, sometimes with "this and that are the same" we mean "up to isomorphism" and so on.

The advice he ends on is extremely solid, though:

  The moral of the story is that we are much better off with the neutral, general standard procedure: name the unknown(s) so that you can formalize all conditions given or to be established and simplify by formula manipulation.

The math will always math.