Why is iCloud.com insisting that I choose a new password?

Hi dave1946uk,

Thanks for using Apple Support Communities. It sounds like you’re being prompted to update your Apple ID password when trying to use the iCloud website. We’re happy to help.

Does the same thing happen when you visit the Apple ID Website?

If so, your password may not meet the current requirements. You can read more under “Use a strong password for your Apple ID” here: Security and your Apple ID - Apple Support

We hope that helps.

Take care.

As you mentioned - But when I try to use icloud.com in a browser (so as to access my Notes on the PC), after inputting email and password and going through double auth verification, I get taken to a screen that insists I "Choose a stronger password".

If PC is a window computer you can't sign in with two factor authentication , if its a Mac see the articles Tips for creating secure passwords on Mac - Apple Support

Learn how passwords are used on Mac - Apple Support

In your iPhone / iPads the Apple ID was signed in with weak password , you didn't signed out Apple ID , however in the browser the weak password will not be accepted as you are signing at www.icloud.com it will prompt to enter strong password ( this is the reason for it ).