Also, there are more things that can be related to "mixing". One of them is called "Silent Payments". Another thing is called cut-through. Because it is possible to send Alice->Bob->Charlie as two on-chain transactions, and have a service, that will batch it, and put Alice->Charlie into the final block. Is it called mixing or not?

It meets the definition, because:

1a. It is advertized as something that can improve your privacy.

1b. It has a mixer function by definition, because the real address is always a mix of what both users provided: the recipient, and the sender. And in case of cut-through, some data from the middle can be removed, or placed as a commitment, by tweaking R-value of a signature.

1c. I wonder if unconfirmed on-chain transaction can be considered a "possibly-transferrable IOU", if it is created in a way, that allows being joined with other transactions, without modifying signatures. For example, is SIGHASH_SINGLE|SIGHASH_ANYONECANPAY considered a "possibly-transferrable IOU"? Because the final transaction can be different than sent by the user in this case.

1d. The transaction can be written with Homomorphic Encryption. Which means, even if it is P2P-based mixing, it is still mixing, right?

2. In case of Lightning Network, it is possible to broadcast the old state of the channel, so yes, "it is possible for the mixer to steal property
passing through it". Even though it may be hard, it is still possible, because you have no guarantee, that the attacker is not a mining pool. And it is not that kind of a bug, which can be "fixed", because you need a pair of unidirectional channels, to prevent stealing (and people won't do that, because that would cause more on-chain transactions, than if one channel can transfer coins in both directions). Which means, even if it will be fixed, it will not be "just a quick fix".

3. No KYC is collected in Lightning Network, Silent Payments or cut-through. This requirement is weird, because that means if it would be collected, it could be allowed. But what if we have Alice->Bob->Charlie, and Bob passed through KYC, but the final on-chain transaction is just Alice->Charlie?

About not banned things: yes, Lightning Network can be treated as an exchange, but it has a mixing function, so is it allowed, or not?