I don't think you're describing quite the same thing. You're talking about forward-mode differentiation, whereas backpropagation corresponds to what's usually c

I don't think you're describing quite the same thing. You're talking about forward-mode differentiation, whereas backpropagation corresponds to what's usually called adjoint-mode differentiation (I think it's called reverse-mode in the post). The difference is in computational efficiency when the number of parameters is large.