Is there a difference between the originally envisioned two way peg sidechain, and the actually deployed liquid federated sidechain?

I think in liquid, ultimately the federation can stop the withdrawals, just like in fedimint.

What exactly are those rules of a sidechain, that would make liquid meaningfully different from fedimint in a custodial context?

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Agree that the two way peg in Liquid is not permissionless. That is what it does not make it a real layer 2 according to the thinking above vs lightning, but this is only difference in semantics.

Re the practical differences between the Elements sidechain and an ecash system it is interesting to thinking that the outcome is quite similar, but there are fundamental differences due to the tech used - blockchain vs ecash.

On Liquid

- the supply can be audited (however more complicated than on bitcoin due to CT)

- need to wait for blocks (1 minute only)

- IBD requirement

- UTXO model (CT hides the amounts)

- and most importantly it is almost prohibitively difficult to start a new Elements sidechain

Ecash mint:

- proof of reserves is not figured out yet (interactive, maybe with key rotations etc)

- fast as only depends on the mint

- better privacy with tokens of fixed denominations which are not connected in any way

- easier to deploy

I am sure there are many other nuances.

My subjective perception is that I would be more comfortable with Liquid for higher amounts or longer time, but for the transactional use ecash seems to be very promising (just like custodial LN wallets have the best UX currently - but without privacy from the custodian).

Spot on.