Dollars prior to 1971 were bearer assets like Cashu tokens. Directly redeemable for the underlying asset (gold) on demand.
Yet dollars prior to 1971 could also be fractionally created, and were. History is littered with bank runs, even in the gold standard era.
And yet, by your definition, the pre-71 dollar was a L2 "currency".
There is no difference with Cashu. It's L2 by your definition..
Enjoy the rest of your weekend.
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A bearer asset is not redeemable for anything. That's the definition of an asset. An asset IS the thing. So, you keep using that term incorrectly.
As far as fractional printing of dollars to gold reserves, yes currency doesn't preclude over-printing. But cashu tokens are not redeemable for the asset like dollars were. They are redeemable for lightning which is THEN redeemable for Bitcoin.
Cashu is more akin to (pre 1971) somone depositing dollars to a bank in exchange for credit to be spent outside of business hours. You can then redeem the credit for dollars, THEN redeem those dollars for gold.
Again, anything above the first layer is not an asset, it is a liability.