Why is it called "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 $$\ce{CO2 + H2O -&gt; CO3^{2-} + 2H+}$$

So you have effectively carbonated the soft drink.

When you depressurize the bottle by opening it, the reverse reaction occurs and you get carbon dioxide.

*Such dissociation is normal and integral to dissolution of polar solutes in water, so generally it is considered as part of the dissolution