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Per the link in the explanation.

"The weight is assigned locally to the router. The value only makes sense to the specific router."



"Unlike the weight attribute, which is only relevant to the local router, local preference is an attribute that routers exchange in the same AS."

Changing the weight on another router will not impact forwarding decisions made by this router.

Local preference is used to prefer an exit point from the AS, It determines which router in an AS sends the desired traffic. Whereas weight is used to select a path on a single router. The question is about routes from local router, hence BGP Weight is the deciding factor.