Hrqls: I agree that in written form they are both ambiguous. English and all other languages are full of sentences that are not clear in meaning. However, I do believe that there should be a way to distinguish between the meanings of those sentences when they are spoken. I am almost sure of my theory about the first sentence --I am an English teacher.--

If you're a teacher who teaches English, you never put the stress on the word 'teacher' when you're saying that sentence. You somehow say the word 'English' more strongly.

I think a similar way should exist with regards to the second example too. But I'm not sure of the details.