I'm afraid that can't be done directly... the web server connects to the database server, that is why the database shows you "system" is logged in and the machine name is the one of the server.
What you could do however is create an activex component that reads out the username and password locally.. But that would require the activex to be installed on the users machine... (nice confusing popups for users)
But as far as my knowledge goes, there is no way to get the username, you might get the ip from the client that connects to the webserver and put that in..
If the page requested is protected via NTLM authentication (see Apache documentation if applicable), you can get the page to create a cookie with the username/ip address/anything else local written to it.
This can then be queried at the server side to get this information.