Question about paint vs dye on LCW.  

Yes
Paint (or lacquer) essentially covers (coats) the wood grain of veneer. The lacquer finish will have a different code on the manufacturing sticker than the dye (and also should be easily discernible by eye).

Thanks for the info, Woodywoo...
Thanks for the info, Woodywood. Is there anything desirable about using paint on these chairs (besides cost)? I'm just curious as to why HM even bothered to use paint on them.

Beats me
I'm really not a fan of painted wood furniture, myself.

Current HM LCW Finishes
From the latest Herman Miller price book...
"This 26 1/2"-high molded plywood chair has natural face veneers, hardwood inner plies, a 5-ply seat and back, and 8-ply legs. When upholstered, the back and seat shells have exposed veneer on their backside.
Note: Red stain (11) shell finish is on birch veneer. This stain allows the wood grain to show through; inner plies are visible. Natural variations in wood grain may affect the appearance of this stain color.
Note: Ebony (EN) shell finish is stain on ash veneer; inner plies are not as visible.
Note: Santos palisander veneer is celebrated for its unique grain and color variations. Similar products ordered in multiples or dissimilar products ordered separately should be expected to have variation in both grain and color characteristics; no two products will look exactly alike."
The following finish codes are in production as of 2013: OU - Walnut CX - Natural Cherry 11 - Red Stain A2 - White Ash EN - Ebony 9N - Santos palisander.

DWR
Shows red lacquer finish as being available.

I thought the painted chairs...
I thought the painted chairs were phased out a couple of years ago.