We're having a terminology confusion issue here, I think.
For there to be two independent clauses, you need each one to have a subject and corresponding verb.
If you have the same subject and do not repeat the subject, you have a compound predicate.
Let's use a simpler example:
1. I brushed my teeth. I went to bed. -- Two independent clauses written as two individual sentences.
2. I brushed my teeth, and I went to bed -- Two independent clauses, joined correct with a conjunction and a comma before the conjunction.
3. *I brushed me teeth, I went to bed. -- Two independent clauses, joined incorrectly with just a comm. (A comma splice)
4. I brushed my teeth and went to bed. One independent clause with a compound predicate.
Now, here's where it gets tricky.
Traditional grammar will say that sentences like 2 require the comma before the and, but in practice it's often left out when the clauses are very short.
Traditional grammar also says that sentences like 4 should NOT have a comma before the "and." However, "Barb's primary rule of punctuation" (since we don't have an Academy of English, I can do this ) says that whatever aids the reader in understanding the sentence is a more important rule to follow than traditional grammar rules.
So while my 4 would not be enhanced with a comma (both of those actions are so simple), a longer passage with a complicated predicate might benefit from one. I talked about punctuation-as-road-signs recently. In that case the comma says "Oh hey! I'm done telling you about the first part of the action, and now I'm moving on to the second part."
(Note that "Barb's even more important rule than that" says that when your sentence gets so long that you consider bending the traditional rules of punctuation, you might look at re-writing to see if you can make it more clear.)
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