How are you logging in and out of the site? Your site should not be functioning that way, which leads me to think that the site is still recognizing you as using an admin  account.

Is it possibly due to incorrect IE security  zone settings? If that's the case then it is possible that IE is automatically logging into sharepoint as the interactive account when the test  account does not have access. If you're testing while logged  into windows with your personal domain account, when the test account hits an area that it doesn't have acccess to IE should be throwing up a prompt indicating you don't have access  to. However, I have seen some errant IE Security zone settings that caused IE to proceed with an automatic login behind the scenes as the interactive user  without the interactive user's knowledge.

To fully validate that the reader  is getting update access, make sure you are logged into windows as your test account, not as your normal, personal domain account.

When I actually went to delete  something it
asked to validate who I was and did not allow the test  user to delete.
Although one of the other admins is able to delete shared  docs from
my MySite. I reckon that is ok with me.