Looks like they've set up an account on your laptop for the domain login. The answer is to set up another user that doesn't log in through the domain, though if you're logging on to their network you'll always have to log in that way (via the domain controller) in order to access it and any other accounts you set up won't be able to access their network or the internet. There's not much you can do about it. Also the content filtering is likely to be running on their server and not on your local machine so this is out of your hands also. Also the "trust relationship" error is appearing because the user account you attempted to log in to the server with wasn't recognised by the domain controller due to it not being on the database.
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