I think that BrotherCreamy would categorize a sidechain such as Liquid, as a 'Bitcoin bank', too. The distinction being the requirement of trust and ultimately who is in control of the funds.
In my opinion (and hopes), banks (or any other similar service) will not be the main way people use Bitcoin. Instead, upper layers should trustlessly provide users with all the required services, whilst ensuring payments can be settled without 'mass exodus on the Blockchain' or anything similar that brings L1 to its knees.

Off-chain networks are not banks. A bank is something with whom you're at to give you your money. You can withdraw your funds all by yourself on LN. The fact that funds are stored in multisignatures doesn't make it centralized, but rather the opposite.