Any elasticity threshold you might come up with to separate one from the other would be completely arbitrary

A dictionary would suffice.

"Plastic" is a general term that covers any polymer-based material. https://en.wikipedia.org/wiki/Plastic

Tires are made of synthetic elastomers which are all polymers. https://en.wikipedia.org/wiki/Synthetic_rubber

Elasticity has no bearing on whether something is a plastic or not, and any attempt to use it as a distinction would be pointlessly arbitrary. Silicone is certainly a polymer...