Article 4 of your list of knowledge points says: When something is the subject, it is better to use start/ to start doing something instead of start/ to start doing something.

Here represents the weather, of course, it belongs to things.

But if it is an informal occasion, such as oral English, there is nothing wrong with using raining. As long as the meaning of the sentence is smooth, people can understand it without learning grammar.

Nowadays, many teaching materials are more inclined to oral English, so the landlord really doesn't have to pursue these details desperately.