What a difference a few letters make - "a" or "the"

A lot of companies claim to be "the" leading provider in this or that. This is a common issue in the payment processing and infrastructure realms. Of course, rarely is there one true "the" and of course you can immediately tell the difference between a start-up claiming "the" and a public company which has to actually defend their claims and therefore more often saying they are "a" leader.
Personally I have always argued for the more mature stance at the various start-ups that I have worked for claiming "a" leadership role in the various realms. Actually nothing is more satisfying than being "the" and everyone actually knows it but you do not have to claim it.
Credibility, quality, execution, etc. really do matter.
So, my advice is to question anyone that claims to be "the" leader in their realm. If they are stretching the truth about that, what else are they stretching?