In this specific case is the proton considered product or catalyst?
As you've illustrated with your figure from the Wikipedia article, a catalyst appears as the same compound on each side of a chemical equation. The short answer to this part of your question is that since the proton only appears on one side of the equation, it cannot be a catalyst. A hydrogen atom bound to a water molecule is not the same thing as a proton. Rather, it is the result, or product, of a chemical reaction. Also note that the reaction given in your question would be more correctly expressed as:  

$$\ce{CO2 + H2O &lt;=&gt; H2CO3 &lt;=&gt; HCO3− + \color{red}{H+}}$$  

In the second part of the question, that regarding the video excerpt, the proton from your reaction actually becomes a reactant in a second completely different reaction. Here, the proton reacts with hemoglobin resulting in two new products: the protonated hemoglobin an an oxygen molecule.  

I hope that I have addressed the crux of your question. Please don't hesitate to ask for any clarifications in the comments below.