I was being stupid 😀
I set up the form correctly, but didn’t test another order after I did that. And thought it wasn’t working, because it didn’t before I checked the enable Run Wsform actions checkbox in Woo tab of form settings.
Today I placed another test order and everything worked as expected: submission is visible!
Only question left is:
how can I also add the billing firstname, billing lastname, billing email address, billing company and order id in the submission record?