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 The argument I hear is it’s adjacent to the lightning network and not on top.

I would argue that you can’t have a mint without running a lightning node.

Curious to see what others think 
 I tend to call Cashu a “permissioned L3” since you still need permission from the custodian to send ecash 
 Not to send. It is bearer held, so you can send it to someone while being offline. Only they need to go online to verify the proofs with the mint. You need permission to swap to lightning Bitcoin.  
 Right. If you can't have a mint without Lightning, it's an application protocol built on Lightning. It's built on top of Lightning. It's a layer on top of an existing second layer. It's layer 3, IMO. But yes, ask calle that invented it 🥹 He knows best.