If it's a bay overo but there is no black showing - all legs, mane, tail are white - then it's not a tri-color but still is a bay Overo or Tobiano. I think it's a perfectly valid term to use to describe that all 3 colors - white, and whatever other 2 colors are appropriate, whether it's bay or buckskin or whatever - are showing.
It also means that neither parent is homozygous for the Agouti gene, which is what turns a black horse into a bay. So, each parent is E?Aa, meaning they have at least one copy of black (the Extension locus, the E) and since they are bay but produced a black horse, they are heterozygous for Agouti - Aa. If either were AA, the foal would have been bay. If one was aa, that one would have been black and not bay.
From paint breeders/trainers/showers
Tri means three meaning buckskin, dun, champagne with pinto markings are all tri colored.