Job titles are not completely fixed by law or custom. Different people and different organisations use different terms. It depends on the culture of the organisation and/or the identity of the person. Also there are fashions in job titles, so sometimes a person or organisation changes a job title.
Sometimes people use all of the above to mean the same thing (except you need to correct the first sentence to say "Person who works at a nursery school or daycare centre.") BUT note that sometimes there are people who work at a nursery school or daycare centre who are not teachers, but perhaps some form of teaching assistant, or maybe a secretary, or a cook,etc.
The best advice is to be as specific as possible. If you mean a teacher in a nursery school then I would choose "Nursery School Teacher"