Pardon me, but I understand double-spending as: spending money more than once. Sure, I can't spend the same input twice as it would violate the protocol rules, but if spend it once, reorg the chain, and re-spend it, it'd be perfectly valid from a protocol point of view.
Yes, you are right.
However, note that by definition, double spending is being able to spend your inputs more than once. Bitcoin uses blockchain which by design doesn't allow that because a single chain can't have the inputs being spent twice. If you were to go by that definition, double spending doesn't exist in Bitcoin. If you were to consider Bitcoin as something with multiple states (which rightfully it can be) and taking it as a system with multiple possible chains, then you can spend the coins multiple times, just not on the same chain.
Now, the definition has evolved into those that we have discussed which can evolve multiple chains, but if you were to consider only a single chain and state, then it doesn't make sense. Only one transaction that spends a specific input can exist in that chain, not two.