What are the reactions occurring during the titration of a carbon dioxide contaminated water sample with diluted sodium hydroxide solution?
Your thinking is in principle correct, let me just add a few more things, that may help you further.

What you are effectively titrating is carbonic acid, or more precise the hydrogen carbonate ion. Carbonic acid is not very stable in water, an most of the carbon dioxide stays the way it is, i.e. dissolved and interaction via van der Waals forces. The main equilibria are as follows (and derivations thereof):
$$\begin{align}
\ce{CO2 + 2H2O &amp;&lt;=&gt; HCO3- + H3O+}\tag1\label{major}\\
\ce{HCO3- + H2O &amp;&lt;=&gt; H2CO3 + OH-}.\tag2\label{minor}\\
\end{align}$$ 
In a contaminated water sample $\eqref{major}$ is by far the more predominant reaction.

When you now add sodium hydroxide solution you shift $\eqref{major}$ to the right side, as you are removing hydrogen carbonate ions via
$$\ce{HCO3- + OH- &lt;=&gt; CO3^2- + H2O}\tag3\label{conversion}$$

As you correctly identified, phenolphtalein is a weak acid by itself and the colour change happens via
$$\ce{H2In + 2OH- &lt;=&gt; In^2- + 2H2O}.\tag4\label{indicator}$$

Once you have added enough sodium hydroxide solution to completely shift the equilibrium $\eqref{conversion}$ to the right, you will start deprotonating the indicator molecule via $\eqref{indicator}$.

It is important to understand, that all these reactions happen at the same time and they keep on happening. This is the basic characteristic of a dynamic equilibrium. Since you are changing the composition of these equilibria the overall composition changes and the system adjusts, see Le Chatelier's principle.