2 of 2 elaboration of why one might use the second formulation.

The alternative hypothesis is β1≠0 OR β2≠0 OR β3≠0, via De Morgan's Laws. Proving any of those three conditions would disprove the null hypothesis.

The breakdown β1≠β2 OR β2≠β3 OR β3≠0 is mathematically equivalent but might be easier to show significance or design an experiment for.