"English teacher" is more common, if you are referring to the person who teaches English. This means "teacher of English language." However, in a different context, "English teacher" could also refer to any teacher who is English. This means "a teacher who comes from England." In this case, the teacher can teach any subject. Let us say you are studying music. And you have one Japanese music teacher and one English music teacher. Which one do you like better, the English teacher or the Japanese teacher? In this context, both teachers teach music but come from different parts of the world. But see the first part of my answer, which answers your actual question.

"English teacher" is correct. "the teacher of the English language" is not technically wrong, but would be used only in very specific situations with much greater specificity.

English teacher or the teacher of English

Both variants are good. The first one is more common. As for the second one it's grammatically correct to use the Definite Article before "English language".