As you analyzed, the three expressions have nearly the same meaning.
Although it is not strict, I will use them properly for the following reasons.
(1) Whether it is "Yamato kotoba / wago" or "Kango".
(2) In case of kango, which is more commonly used, or whether there are many homonyms in a particular case.
It is essential to pay attention to the number of homonyms, especially when used in conversation in order to avoid confusion.
So, when you are talking about school, since honkoh has fewer homonyms than tohkoh, in general [本校]{honkoh} is safely used in conversation.
As for Yamato kotoba / wago or Kango, I'll show you some useful infomation from here.
It says:
Yamato kotoba (大和言葉, literally "Japanese words") are native Japanese words, meaning those words in Japanese that have been inherited from Old Japanese, rather than being borrowed at some stage. They are also known as wago (和語). Together with kango (漢語) and gairaigo (外来語), they form one of the three main sources of Japanese words (there is also elaborate Japanese sound symbolism, of mimetic origin).
Very roughly, kango are generally more formal, often restricted to writing, while yamato kotoba are more casual and more often used in speech, but both types of words are commonly used in both speech and writing.