> [...] the merchant can send an invoice for a payment. Either the customer manually pushes the payment, or delegates to the bank that each invoice from merchant X should immediately result in a payment push [1].

This already very close to how SEPA direct debits currently operate. I can instruct my bank with one click to stop honoring a given direct debit mandate (they'll then block all further payments under the same mandate reference), request any payment to be reversed for any reason (that I don't have to provide) etc.

The only difference to your suggested model is that the default is to honor all new mandates. I believe nothing – operationally or from a scheme perspective – prevents banks from requiring positive confirmation for every new mandate or even every single direct debit, though, and some banks (but not mine) even support this.

> in the pull system the customer can only beg the merchant (eg. the gym) to stop charging their account.

Not for SEPA direct debits, in any case.