When setting up the Engine DSN, you MUST provide the Admin login name. In some cases, you may need to provide the full context as well, as in ".Admin.MyOrganization" to sign in properly. Note the leading dot, which is also important.
Once the Engine DSN is created (which is only done once), then anyone (including standard users) can create a Client DSN on their workstation that points to the Engine DSN that you created. Simply specify the Server name, then click on Get DSN List, and pick the one from the list that you want.