gwern comments on Harry Potter and the Methods of Rationality discussion thread, part 15, chapter 84 - Less Wrong
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In Methods of Rationality, the Weasley twins refer to the Map as being part of the Hogwarts security system. So it probably gets the information from the wards.
My interpretation of the quote is that the Headmaster overrode the usual process of identification (which is automatic) in order to protect Quirrell's privacy; in that case, the Map would also know nothing beyond "The Defense Professor".
An alternative interpretation, however, is that the circle-drawing bit was only meant to key Quirrell into the wards as a professor. Normally, I suppose, this would be done by something like "I, Headmaster of Hogwarts, declare Quirinus Quirrell to be the Professor of Defense Against the Dark Arts! So mote it be!" In this case, Quirinus Quirrell was a false name, and Dumbledore knew this, so he used an alternative process that doesn't require a name.
Yes; remember that they speculate it was made by Salazar himself, and remember in ch43 Quirrel independently says:
The twins never mention in canon seeing the Basilisk on the map.
(Salazar seems to have specialized in security - one of his other talents was being a Legilimens.)
That would have borked the plot, and it's entirely probable that Rowling hadn't even come up with the Marauder's Map by then.
In MoR canon it probably makes more sense if it doesn't show up on it though; the whole Chamber of Secrets certainly doesn't.