> ...if two or more people strongly disagree about facts, is to drill down into the underlying definitions that provide context for the facts.

Except that surely you don't have the time to do that most of the time? What about cases where the other person is being dishonest and shifting the burden of proof about a claim, or trying to offer more and more arguments which detract from making the original point?

While drilling down is possibly indeed what you should do, it's also easy for it to derail any average discussion, if done wrongly and either of the participants don't have honest intentions of having a discussion and finding out the truth.

Indeed. Many philosophers are pretty clear about it (French ones: B. Pascal, A. Lalande...), however it implies cold (dispassionate) heads and (as pointed out here by KronisLV) plenty of time. Most of us, sadly, lack time and only invest it in whatever we feel some passion for.

I have heard that philosophers today have a culture of high toleration for very heated discussions that would be considered beyond appropriate in other contexts.