Comma, Comma, Comma, Comma-chameleon? A comma joke! English nerds unite!
I actually don’t like the use-the-comma-when-there’s-a-pause rule. It’s very widely used, but it too often leads to unnecessary, incorrect and potentially confusing commas in the middle of very long sentences.
My pet peeve is seeing random commas between the subject and verb of a sentence because the writer felt like the reader needed to take a breath. For example, “The tall green apple tree at the foot of the hill at my grandfathers farm, is dying.” Even more common, people tend to put a comma before a conjunction in a simple sentence when there are two verbs. For example, “The tall green apple tree is at the foot of the hill, and grows badly in the partial shade.”
As you explained (very well, I thought) commas set off dependent clauses and separate independent clauses from each other. I think a good rule of thumb for comma usage is that, unless commas are used in a list, a date, or an address, you should be able to remove the part of the sentence set off by commas and still have a complete sentence. For example, “The apple tree, which my grandfather planted at the bottom of the hill, is dying” makes sense if you take out the part offset by commas and just have “The apple tree is dying.”
When commas are used correctly, they are a great way to help the reader break a sentence up into digestible blocks and figure out which parts are the most important. I think American schools often do a disservice by teaching that commas are stage directions instead of meaningful structural elements.