I believe that you are correct in referring to the covering on the ceiling of the car as the headliner, with the exception of that sliding piece. I had been referring to it as a sunshade, but that term apparently refers to the curtain that moves up and down inside the rear window, rather than back and forth underneath the moonroof.

The panel underneath the glass moonroof actually comes out (rather easily) through the roof itself (at least on my patient, a 1996 C280). From inside the car: manually slide panel back, electrically pop sunroof up to vent position, pull bottom edge of rubber bellows out of slides they are attached to, and pull the slides inward (toward center of car) to detach them from whatever it is that's gripping them (they are detached from the car at this point). Label these slides so that you know left, front, top, etc. Then electrically slide the moonroof back. From either inside or outside the car, pull panel forward and out towards the front of the car.

On mine you then have to then pry out two plastic gills, and the plastic handle, without breaking them. You then have a small version of what you've dealt with before.

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1996 C280 289K Traded
1997 E420 167K Traded
2001 S430 240K Traded
2010 E550 4matic 80K
2000 GMC Jimmy 198K Gone to Boneyard
2003 Camry LE 196K
2011 Mazda3 i Sport 31K