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@DeaneYang I somehow agree with your argument. One thing that is not clear enough (at least to me) is the well-definedness of (2), since a section of the pull-back bundle may admit different linear combinations. I believe we should prove that if $F^*(s)(x_0) = 0$ then $F^*\nabla(F^*s)(x_0)$ should also be zero. However I cannot prove this...
If we define by this, then well-definedness problem comes out: for any section, it may admit different linear combinations. So how to prove the definition is independent of choice?
