Are the Mac OS X GUI and a Terminal session peers?

"An expert can feel free to correct me here!

s001 is your TTY. It's the terminal you're connected to the machine via.
The GUI is, as you said, a means of logging on to the server, and therefore it shows as console. It does have stdin, stdout, and stderr
stdin is typically NULL. stdout and stderr go to the console log, in /Library/Logs/Console/*your_uid*/console.log"