That rule isn’t universal though. “I go to the dentist every year” doesn’t imply that I particularly care about a specific dentist (or, case in point, about the specific dentist).

“This was a nice evening” is also difficult to characterize as not caring about which evening it is. As are books titled “An Introduction to XYZ”, where both the authors and the readers will probably care about the specific book.

This doesn't touch place names. The Hague, Amsterdam, The Maldives, Australia, The Netherlands, France...

Nope! Because "the" also refers to a class/category which is essentially the opposite of indicating a particular member.

For instance, "the bald eagle is native to North America".

There is also "The" in names; The Beatles, The Great Lakes, The Bahamas, The Bronx, ...

The role of "the" depends on context: how was the subject treated in previous sentences in the conversation.

"I see a bald eagle out of the window. The bald eagle is huge!"

In this second sentence, there is no way that "the bald eagle" could refer to the class of bald eagles, because the previous sentence introduces the topic of the sighting of a bald eagle. It refers to that one.

If "the bald eagle is huge" appears out of the blue, not preceded by any discussion of any eagle, that tends to bring in the class interpretation.

I learned the above through Japeanese. A similar thing happens with the は (wa) topic particle; I realized that "the" works in a similar way in that regard.

That much is obvious and everyone learns it like on the first week of English learning, of course. It’s the various cases where that simple rule doesn’t apply that are difficult.

"I play the piano"

And then there are those cases where you actually use neither!