The word is far older than the first Latin mention. That's for sure. It goes back to the legend of "Father Czech" the leader that lead them to their new home (Bohemian basin). Bohemia comes from the old celtic tribe that wandered out centuries before, from Boii to Bohemia (same etymology as Bavaria perhaps). When exactly Czechs started to call themselves that is hard to say because it's older than their written history.

Since it was part of the Holy Roman Empire since it's very early days (11th century or so) the German "Böhmen" gained traction over the Slavik "čech". To this day it's still weird to spell in German --- "Tschechisch", is what the language is called in German. Czechs are just trying to get the rest of the world to call them what they've always (for a millennium) called themselves.