I get your point, and you can pretty much tell it is a Catholic school from the name, but in California, there are plenty of public schools named after Catholic saints because there are a ton of cities named after Catholic saints. Santa Monica High School, San Jose High School, San Miguel Elementary School. I think the tip off is whether the it's anglicized or not.

The tipoff is whether the schools' name is based on the name of the city, no?

In the Los Altos Mountain View area, I can name St. Nicholas, St. Simon, and St. Francis which are all private schools, none of which are named after the city that they are in.

There are lots of schools named directly or indirectly after saints that aren't also in cities named after saints.

> I think the tip off is whether the it's anglicized or not

There are definitely both public and private schools in California with the names of Catholic saints in both English and Spanish on each side, so that's not a certain tip off.