Vlad Thanks. That's fine for me.
I may have taken this issue a bit too far, and got carried away. Apologies.
In an effort to say something useful still: I may have to put partially the fault on me as I am using the Privacy Badger browser extension which did not allow your page to properly display some details.
Then, I would still argue that it is a bit confusing for a user to be shown a payment confirmation page just immediately after entering its user and password for the first time. If I had gotten some kind of previous welcome page explaining that the next step was going to be interacting with a third party site, I would not find anything confusing, they fail all the time and I know how to work around them.
Note: I tried again and it's not Privacy Badger's fault, it seems there was some kind of persistent connection problem with Stripe for me that led to partially loaded forms. You were right that Stripe was using so called Links, but I could not see all details about it yesterday. Today after reseting the Links account I can finally see the forms on your site as you guys were showing to me. Thanks.