WebService not set correctly

Update: restarting the editor appears to have fixed the translate button, so that's good.

Great!!

But I'd still like to know how to move the app to my work Gmail

Its not enough to CLOSE the tabs in your browser, because internally the account is still logged in. Thats why when you reopen your browser, you don't have to type the password or select the user.

You need to SIGN OUT of both accounts by clicking the sign out button that shows when you click your photo in gmail.
Then, after you signed out both account, log in your work account. (only that one)

That will ensure that your primary account is your work one.
When the plugin installs the webService it just creates a copy of the WebService file into the account that its logged in your browser (or the primary account if you have several logged at the same time)

Another way to double-check your primary account in your browser is the work is by checking that its number 0:
e.g. mail.google.com/mail/u/0
instead of mail.google.com/mail/u/1

Hope that helps,
Frank