SOLVED!
After restarting and relaunching Designer again, the splash screen was displayed again.
This time (I did not tried that before) I just hit the far right column (BUY) again.. (I could always cancel the payment confirmation, right?)
Then... it suddenly worked: the screen (that apparently should have come up the first time I bought it). without a second payment, of course. Now I could complete the process and link my Affinity Serif account successfully. So, it apparently checks the previous purchase status before submitting a new purchase. (something one expects when you click that small link 'restore purchase' on the bottom.... hint for v2.0.1)
In short: just smash that 'PURCHASE' button again! (after reboot)