Up-and-down head movement is what's bad. It shifts your spine angle, and causes you to lose balance - your swing falls apart.
Lateral head movement is OK, as long as you swing smoohtly. Allowing lateral movement helps with weight shift on takeaway. Also, too rigid a head can choke off your follow-through on your downswing.
I know that iacas worries that head movement can be the start of things going wrong, as did Nicklaus. A still-head proponent, Nicklaus wrote that his homecourse pro Jack Grout used to hold onto a handful of his hair to teach him to keep his head still.
For Hogan's Five Lessons view, I spend 20 minutes reviewing the book. The only mention I found of the head was that the chin should be in contact with the left shoulder at the top of the backswing. Didn't mention head activity, unless I missed something.
This GolfLink piece says mainly that Nicklaus as a player (on his drives) and Tiger Woods had lateral head movement. This piece says the head needs to remain behind the ball at impact.
Jim Flick also mentions Tiger's movement, and suggests that people confuse a little head movement during the takeaway with major swaying:
So, what is the nature of still head? Possibly, are some swing styles tolerant of lateral head movement, whereas others need the still head?