The same blog post you mention says:
The definite article the is used in front of any noun the listener or reader already knows about.
So, when you say:
Our teacher said that we need to use articles.
we now know about the teacher. Therefore:
The teacher talked about differences between two types of articles.
Is the natural way to say it, because the teacher was mentioned in the previous sentence. You wouldn’t use a instead of the, because you are talking about the teacher you already mentioned.
You could word something this way, though:
A teacher said that we need to use articles. That teacher talked about differences between two types of articles.
By the way, the “rule” you initially cited is trying to say that you don’t use the word teacher without any kind of modifier. In other words, you would NOT say this:
Teacher said that we need to use articles. Teacher talked about differences between two types of articles. (incorrect)
We don’t generally begin sentences with words like teacher, carpenter, doctor, or policewoman. Instead, we use article, possessive pronouns, or some other determiner:
Our teacher said...
A carpenter told me...
My doctor wants...
That policewoman saw...
Best Answer
“Student”, “policeman”, and “teacher” are singular count nouns describing what someone is or what job they do. We use the indefinite article in such situations. E.g. “Are you a student?”, etc.