Step by step to restore Time Machine b/u (or original internal ssd) to new bootable drive also updating OS

• Install the desired newer OS onto the external drive

On modern macOS, You MUST install macOS First, and ONLY from fundamental sources. In general, "cloned" or "copied" macOS will NOT RUN, due to the crypto locked virtual System Volume.

>> So install the macOS you want onto the Internal drive, and use it to set up with NAMES for both accounts:

Use the Time Machine backup to Migrate ONE account onto the Internal drive, then create (but do not import data yet) for the second account.

next, you want the second Account's HOME folder (-OR- just its major library folders) to be on the external drive. You can change the default location of the Home folder for that user, and everything inside it to a different drive.

-OR-you can do the latter (just large libraries) easily with Aliases (pointers that transparently link to a new directory, even a different drive)

Here are some third-party articles that discusses the subject:

https://www.lifewire.com/move-macs-home-folder-new-location-2260157

https://appleinsider.com/inside/macos-ventura/tips/how-to-move-your-home-directory-in-macos-ventura

MUCH more information is available, just ask.

There is no such requirement except that the account database (account name and password and home folder location) be on the boot drive.

your files can be moved to any connected drive, and major libraries like photos and music can be moved to any connected drive.

You CAN reboot the machine if you prefer, but that is not necessary.

With two accounts on the same macOS version, you are literally running the same instance of MacOS for each Account, possibly even simultaneously, or you can log out as one and log in as the other if you prefer.

Before you begin, I recommend creating a NEW Admin Account. Think of this as your Administration-Only account, and don't store a lot of stuff there. Write the password down on a pice of paper and stash in a safe place. This is your ticket to rescuing your Mac if you mess things up later.

DO NOT CHANGE the name of a Mac account yourself manually. One mistake and your Mac is un-bootable. if you MUST, create a NEW account with the name you want, then COPY all the files to the new account. After a week, if you did not re-visit the old account, it can be deleted.

When Migrating you old user accounts, you can get Migration Assistant to slightly modify the username of one or both accounts as it is Migrated. You have the option of Duplicating accounts (Migration assistant will change the name of the incoming account slightly) or Merging accounts (Migration assistant will merge the old files into the existing account with the same name.)

To avoid duplication, consider diversifying into an account with only your first name, your first name Plus Initial, Mr.YourSurname, a nickname, 'your name-work', 'your name-ABC corp', yourname-personal, and names like that.

To keep things separate and avoid mistakes, I recommend a different password for each.

<< unless there's a way to fast-switch accounts between the two without restarting. >>

yes, that is called fast user switching

Switch quickly between users on Mac - Apple Support (TM)