How come, if you have multiple users, you do not first get the list of all the User Accounts to choose from? How does it know which one to start with from cold?
I believe the OP had XP installed as a single user, without password, and system would boot up into Desktop. The install of Logmein could have split User and Adinistrator. thus causing RunAs window.
With multiple users the Logon is a shortcut to program, and is why I suggested the Shortcut Fix previously. As the RunAs appears at Logon it maybe reason does not appear with other programs
I donot like to mess with Registry but it could fix problem.
Like all windows errors, it is a case of try and see.