OC's point is that the chain rule for partial derivatives shouldn't be assumed because the ordinary chain rule holds, there's more depth to it than that, and the proof is harder than you might instinctively expect based on the ordinary chain rule.
It's epistemically acceptable to understand these both as "the chain rule" once we're satisfied they've both been proved, and apply liberal amounts of synecdoche from there (and I don't think OC disagrees with you on that).
Actually by 'ordinary chain rule' I am referring to what you're referring to as 'the chain rule for partial derivatives'. It seems like backprop follows very quickly even from that, but it does not.
I agree. That's what I'm referring to as 'the ordinary chain rule'.
> so it's still technically just chain rule
No. Go try to derive backprop for general DAGs using only the chain rule. If you complete the proof, then you will agree that the proof was more elaborate than you ever expected.
It's epistemically acceptable to understand these both as "the chain rule" once we're satisfied they've both been proved, and apply liberal amounts of synecdoche from there (and I don't think OC disagrees with you on that).