I have two questions for you:
These two sentences mean the same, don't they?
I am an English teacher. You could be either a teacher who is English or a teacher of English - context would establish which.
I am a teacher of English.
Are there any differences between them, does the first one imply that the teacher in question is English?
When you finish the Faculty of Philology,with the English language as your main subject, your title is -a teacher of English(the English language)-, right?
There is no title attached to merely completing a course. You will receive a degree which may say that you are now a BA [Bachelor of Arts] or something similar. To be called a teacher, you will need either a qualification in teaching or to hold a teaching post in an institution.
What about professor?
As far as I know, it is not a title, it just means that you teach at the college or university.
Is that right? A professor in the UK is someone who is a teacher with the highest rank in a university [usually the head of a department]; in the US, it is more broadly used for any teacher of high rank.