Everyone calls it MEV because of this very paper. They coined the term. You did answer your own question in (3).
Also in this context, the reason that these sandwich attacks can happen is because all the information is public. The adversary can see your transaction waiting to be validated, and pay the miner a higher gas fee to be executed first.
Index front running is not illegal and is based on public information. So, this may be a form of legal front running.
Most likely none of the transaction signers qualify as clients of the miner. Transactions are transmitted over the network anonymously via a gossip protocol, and hundreds or thousands of miners have the chance to include (or not include) any transaction in a block. Transactions are selected for inclusion in a block effectively randomly, through an entirely mechanical process, and no relationship is established or maintained between the miner and any transaction sender.
In order to assert that a transaction signer is a client of the miner that builds the block that includes the transaction, you would have to redefine what the word “client” means.
This does not seem obvious to me; even if "client" is too strong a word, the transaction signer and miner have some social contract that's very similar to more traditional fiduciary duty, even if the technical details and enforcement mechanisms are totally different.
The primary rationale of fiduciary duty is trust. In contrast, the whole reason that miners even exist is so that the service they provide can be performed in an entirely antagonistic environment, without trust.
I suppose it can be said that a miner’s “social contract is very similar to more traditional fiduciary duty”, but only in the sense that a thing is somehow conceptually related to the exact opposite of that thing.
Miners are anti-fiduciaries.
This is a novel construct. It may be wise. But it’s not the status quo.
Words have meaning relative to their context.
The usage is popularized because it makes sensationalist headlines and forum headers, that the "generalized usage" was either misguided or intended to confuse, which correlates poorly with accurately conveying the meaning of word or capturing the details of the blockchain phenomenon.
No, not because of vague connections, but because it does the same kind of destructive thing that’s bad (for the ecosystem) for the same reason it’s bad on the normal market: it extracts value from a transaction because of timing and a privileged position, disencentivizing positive-sum behavior. (In the broker’s case, because of knowing about the transaction; in the miner’s, because of having that mining power and being able to quickly execute on knowledge of upcoming transactions.)
Trading fast based on "slightly obfuscated" public domain information is categorically not any kind of front-running. It is as far as I can see just a pure execution arbitrage.