Can someone try to create a new account at https://centralmarket.turnstilesystems.com/Account.aspx?returnUrl=/MyAccount.aspx and see whether 1P will prompt to save the new login/password?
It didn't for me
By the way, I just changed my password on appleid.apple.com, and 1P didn't prompt me to save the change, either
I've tested both and been able to replicate your findings in each case. While the cooking school website didn't prompt me to save a new Login on the registration form it did prompt me when I went to next sign in.
More annoying, at least in my opinion (and you may well disagree) is not recognising the password change form on Apple's site as it certainly meets the criteria as far as I can see, your original password and the new password twice.
So we have some work to do here. I'm not sure if you're aware or not but when you use the Password Generator to create a new password for you it will automatically create a new Password item whenever you click the Fill or Copy button. The purpose is a safety net given us mere humans are never going to remember one of the passwords generated by 1Password. The hopeful workflow is 1Password creates this Password item and then after 1Password has correctly identified that it should ask if you want to create or update a Login item. Once the Login item is saved/updated the Password item is removed as its purpose has been served and the password is being correctly stored in the right place. When this workflow breaks down though for whatever reason e.g. 1Password doesn't correctly identify that it should ask if you want to save, you still have the password item accessible. If you look in the Password category you should find two Password items, one for Apple and one for the cooking school. In both cases I found that attempting to log in and using the password item to fill in the password resulted in 1Password asking if I wanted to save.
Hopefully the safety net procedure described will help you until we can determine the causes and get a fix in place. Thank you for reporting both of these as I personally wasn't aware of either.
ref: OPX-1009 & ref: OPX-1114