"Summer", like the other season names, is often treated as a name (and sometimes written with a capital letter, like other names). Examples 1 and 2 do this.
Example 3 is not a counter-example: "summer" is there used as a modifier for "solstice", which is a common noun and requires an article.
Example 4 is a bit more complicated. "Song of Summer" would be possible, but would suggest Summers in general; "Song of the Summer", especially since a year is given, suggests that particular Summer.
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
A better statement of the definite article "rule" would be that you should use the X only when there is a reasonable expectation that your readers or hearers will understand what particular X is meant.
This includes situations when you have previously mentioned X; but it is not necessary to have mentioned a specific train, because few people hang around a railroad station with the plan of taking "a" train, any train that shows up. They're there to catch (or meet) a particular train. Even on a subway system where trains run on any given line every fifteen or twenty minutes, people are waiting for the next train.
Consequently, the hearer or reader may be expected to infer from context that what is meant is “the” train the speaker/writer is waiting for: the train to or from a specific place scheduled to arrive here at this specific time.