Teach me how Catholic school funding works

People talk about Catholic school funding as though everyone is paying for it out of their own taxes. Most recent example I've seen is from this article, written by a Catholic school parent:

Now, you’ll hear defenders of the current discriminatory system say that the Catholic board is funded only by Catholic taxpayers who choose to direct their school taxes to the separate system. Not true: Catholic and public schools alike are funded by general provincial revenues, and the amount of funding each school gets is determined for both systems by a per-pupil formula set by the province. But even if it were true, it would be grossly unjust. Those who send their children to private schools aren’t exempt from paying taxes to fund the public system, just as those who drive are not exempted from paying taxes to support transit capital costs.

So here's where I'm confused: from time to time, I receive a form from the municipal property assessment people that contains a question about which school board I want to support. I don't remember the exact wording and I don't have a form with me so I'm probably missing some information, but I remember that anyone can support the English-language public board, people who meet certain Catholic requirements (which I meet by virtue of having been baptised) can support the Catholic boards, and people who meet certain Francophone requirements (which I do not meet) can support the French-language boards. You can vote for trustees only in the board you support.

So is everyone really paying for the Catholic boards, or are they only being paid for by people who direct their taxes that way on their assessment? If they're being paid for by everyone, why does the municipal property assessment ask us which board we want to support? If they're only being paid for by people who direct their taxes that way, why do so many people think they're being paid for by everyone?