And, I have tried multiple accounts from local to Microsoft live accts. I can't see any difference in the way the users are defined on this new computer than on my others--one with 8.1 already, and the other with win 7. What am I missing here?

1. What troubleshooting steps did you try to fix the issue?
2. Which antivirus program is installed on your computer?

Method 1: I would suggest you to reset Windows store and see if that helps.
a. Press Windows key + Q.
b. In the Search box type WSReset.exe and hit enter.

This will reset your cache for the store. Now try the download and check.

Method 2: If the issue still persists, then I would suggest you to enable built-in administrator account and see if that helps.
a. Enabled the "administrator" account.
b. Logged in as administrator and went into User Accounts and managed the original account I want to use to download Windows 8.1. I changed account type from Administrator to Standard. Then changed account type back to Administrator. c. Logged back in as original account and now you will be able to download 8.1.

To enable the administrator account you have to get an "elevated command" prompt, then type:

$ net user administrator /active:yes

To get the elevated command prompt, you can bring up task manager, click File, then hold the control key and select "Run new task".