Did you try changing your userinfo URL? LAMS won't create a new user for the person coming from Moodle if your userinfo URL is incorrect. You need to replace '%root%' with '%username%'.
Your own Moodle user can access LAMS probably because it already has it's own account created in LAMS - <username> prefixed by 'mdl_'. At some point the userinfo URL was probably correct when this account got created.
Incidentally, this also means that when creating different integrated server entries, the prefix value also needs to be unique. I think this is now enforced in the database, but not sure. This comes into play for example if the built in admin user in Moodle with username 'admin' tries to login to LAMS - if there are multiple Moodles with the same prefix, then they both get recognised as 'mdl_admin'.
Posted by Jun-Dir Liew