Homonym vs Homophone

Words that sound alike are homophones. (not homonyms). (this is purely my definition), especially in the sentence with words like: 'They're/their/there'.

Oxford (online): 
Each of two or more words having the same spelling or pronunciation but different meanings and origins.
Example: For example, pole and pole 

Ok then, I know this might incite some debates, but the definition in the Webster:

Homonym can be troublesome because it may refer to three distinct classes of words. Homonyms may be words with identical pronunciations but different spellings and meanings, such as to, too, and two. Or they may be words with both identical pronunciations and identical spellings but different meanings, such as quail (the bird) and quail (to cringe). Finally, they may be words that are spelled alike but are different in pronunciation and meaning, such as the bow of a ship and bow that shoots arrows. The first and second types are sometimes called homophones, and the second and third types are sometimes called homographs--which makes naming the second type a bit confusing. Some language scholars prefer to limit homonym to the third type.

Why don't we just define that homonyms are strictly words with identical pronunciations and spellings but different meanings?

We have these 'words':

1.Homophone: Each of two or more words having the same pronunciation but different meanings, origins, or spelling.

For example, new and knew

2.Homograph:  Each of two or more words spelled the same but not necessarily pronounced the same and having different meanings and origins.

For example, bow and bow

So, why don't we make use of this?

Source:  https://www.espressoenglish.net/13-grammar-mistakes-that-even-native-english-speakers-make/