state schools vs. public schools
Beware of the word 'public school' in British English!
Types of schools in the UK: http://projectbritain.com/education/schools.html
As you can read there, if you work at a CEIP/IES in Spain, you would probably describe it as a 'state school'.
A 'Public school' in the UK is a rather expensive type of private/independent school (Sorry, it's a false friend for Spanish speakers). Royalty, future politicians, children from very affluent (rich) backgrounds tend to attend these schools.
If you want to refer to Catholic schools you may use the term 'faith schools' (That would include all faiths, not just Christian).
Types of schools in the USA: https://www.readingrockets.org/article/choosing-school-types-schools
It is true that Americans refer to state-funded schools as 'public schools'. My advice, anyway, to avoid confusion, would be to refer to a CEIP or IES a 'state school' or 'state-funded school'.