I do not have multiple accounts so I can't reproduce what you are seeing. Conceivably it is showing both because you are using one account for the app store and the other for everything else. If you are signed into the same account for both then conceivably if you are still seeing another, it is perhaps detecting the presence of apps from another account? Again, I don't have that on my devices so I am guessing.
The whole system is not designed to accommodate multi-region use. If anything, that could even be read as violating Apple's terms of use, so I suspect trying to do that is made very difficult by Apple. You are trying to do something that devices are not configured to accommodate, so don't be surprised if it is not working.
You can only have one primary email address. That one is your Apple ID that you use when signing in. Apple lets you add other addresses as a rescue address.
Typically you entirely erase a device between signing in with separate accounts. If you do not do that you end up with possibly merging data from the two accounts.
Apple's change from "Apple ID" to "Apple Account" was a change in naming. An Apple ID used to refer to both the email address you used and to the entire account, so that was confusing. Nothing else changed except how things were called.
There is no simple solution to multi-region use of devices. The only one I know of is to buy a second device that you use when you need to use apps from another region. Even then you may have problems in configuring it if you do not have a mailing and billing address in the other country too. You do not necessarily need phone service on the other device if you can make it work over the Internet alone.