>>The user has to be set up distinctly. For example, I would have to set up MYDOMAIN\colem with a login on the server, and then give that login specific rights in the database.
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>In my case, this is a connection string that is defined in a configuration file of the Web site. The reason that it is working, could it be that this is because the Web site has some kind of setting somewhere that it would use a specific user which would have rights to access the database with Integrated Security to Yes?

Could be. I would check the logins on the server to see which Integrated users exist. The website could be running either under the system account or could be impersonating a user (maybe you).