I was quite shocked when in Strasbourg I read a plaque that said that something had been built by the 'culte catholique', but from the dictionary definition I cannot tell whether it is a polite synonym for religion. I'd appreciate an informed opinion.
This is actually part of a famous pair of false friends: What's usually referred to as a cult in English is called a secte in French, while a culte in French is closer to the English worship or to rite, when it means the public practice of a ritualised set of religious acts.
Secte has kept its technical meaning of subdivision of a religious or an intellectual movement in scientific writing, similarly to its English cognate, but in everyday usage the cult meaning predominates.
Culte has no negative connotation in French.
In a religious context, it just means a belief. A lieu de culte is used to name any place of worship like a church, a mosque, a synagogue, a temple, etc.
Otherwise, culte is often used in apposition to qualify a movie (un film culte), a series or a catchphrase (une réplique culte). In this latter case, culte always has a positive connotation.