Exactly. In the vanilla case, the client sends the payment, waits for transaction confirmation, obtains the transaction ID, and only then sends the request. Even on L2 this may take seconds. With preconfirmations (the way I understand them), the L2 operator (sequencer) can issue a signed promise of transaction inclusion before the block is even produced. If the server is willing to trust those promises (which may be crypto-economically backed, cf. shared sequencing with restaking), the server can deliver the service right away upon receiving the request + payment preconfirmation.
how we define atomicity in this environment. What would be an ideal execution in this sense?
Ideally, we may want to see some cryptographic (or, perhaps, cryptoeconomic) mechanism such that the paymend goes through if and only if the service is provided. This is indeed tricky when dealing with information transfer. I don’t have a concrete solution, and it may well be the case that in the Lightpush example atomicity is infeasible. It may be feasible though for other, more cryptographically self-contained services (within our future generalized service marketplace) - something along the lines of “I will pay you for a preimage of this hash”.