With ligthning, you're not using the bitcoin network to process transactions. If we opened a lightning channel for our 2 personal bitcoin wallets, and I paid you 1000SAT over the channel, and then someone checked my bitcoin wallet, they would see that I still have all those SATs (of course, I can't double spend them).
Did Jo and Alice participate in a USD transaction?
If Bitcoin's original thesis of truly decentralized P2P cash still holds, then it should be viewed as an M0 money supply. Financial services and products build on top of and separately from the main chain make sense as analogues to reserve banks and payment processors that do not handle the majority of transfers in physical cash.
Calling it a promise makes it sound like there's counterparty risk and you're relying on them not changing the code. Changing the lightning code would have no effect on your funds locked into lightning using the old code.
Similarly, if I send you 1000SAT through the lightning network, I am trading in bitcoin, but not using the bitcoin blockchain to do so.
Sure, Lightning relies on guarantees offered by the bitcoin blockchain to promise payment security. But it's not the same thing, and the Lightning network could become compromised even if the bitcoin blockchain remain secure For example, if Lightning accidentally started adding an extra 0 to transactions, if I send you 100 SAT over the LN you could see that you are receiving 1000 SAT and send me some products. Later, when the channel is closed, the bitcoin blockchain would refuse the final transfer, but it may be too late.