I think you may be confusing "final approach segment" vs. "final approach fix." To the best of my knowledge, an ILS does not have a FAF, per se. That is depicted on ILS plates as a LOC procedure. The final approach segment on an ILS begins where the GS intercepts the lowest published altitude on the intermediate segment, or if given a lower altitude by ATC, where the GS intercepts the altitude given.
As a practical matter, on most approaches the FAF and GSI are at the same point. I see the issue coming into application if you were shooting an ILS under part 121 and the reported weather went below minimums.
As a practical matter, on most approaches the FAF and GSI are at the same point. I see the issue coming into application if you were shooting an ILS under part 121 and the reported weather went below minimums.