Unable to change Apple ID password because I was unable to "continue on one of your devices"

I am trying to change my Apple ID password.

I logged in at https://appleid.apple.com/account/manage and started the change password process.

However, after entering my existing and new passwords and pressing the 'Change password' button, it presents me with the following instruction:

This instruction is too vague for me to execute.

I tried going to both my iPhone and iPad and nothing unusual happens. I was expecting some kind of authorization process to be initiated automatically.

If I go into settings -> Apple ID, iCloud, ... on these devices I see nothing that I can "continue" there.

Does "continue" mean "sign out of iCloud on your other devices"?

(But I checked the box 'Sign out of Apple devices and websites associated with your Apple ID' on the previous dialog so I don't see why I should have to sign out of them myself.)

Can someone please explain how to "continue" and complete the password change. I cannot get beyond this step and my password change is not implemented.