Why is it called “carbonation”?

Why is it referred to as "carbonation" and we drink "carbonated" beverages when carbonate is $\ce{CO3}$ while $\ce{CO2}$ (carbonite?) is present in carbonation?

Nope. Carbonate is $\ce{CO}_3^\color{red}{2-}$ (an ion), not $\ce{CO3}$. Carbonite is $\ce{CO}_2^\color{red}{2-}$.
Carbonation involves dissolving $\ce{CO2}$ gas in water. It turns out that $\ce{CO2}$ reacts (maybe not the best term to use here*) with water via the following reaction $$\ce{CO2 + H2O -> CO3^{2-} + 2H+}$$