BTW the full name includes the definite article, so it is "the Czech Republic".
Funnily enough the distinction between regions exists in english language - Bohemia being the name of the region. So Czechia as name for country could work. But in czech language word for Bohemia region is "Czechia" (there is no Bohemia).
So thats why for years you had people insisting on the Czech Republic. Because you don't want to overlook the other two regions Moravia and Silesia.
No. Bohemia is Čechy, Czechia is Česko.
Yes, they are sometimes confused, and maybe people are unhappy with Česko because it's just too similar to Čechy. People from Moravia and Silesia feel underrepresented when someone mistakenly uses "Čechy" (Bohemia) for the entire Česko (Czechia; Bohemia+Moravia+Silesia).