Welcome Ernie. I've never seen one with a lap deck lid but I'm no expert. Someone will be along soon to give you a knowledgeable answer.

Thank you

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Thank you

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I don't own a '31 but there was an article in the G & D some years ago about some differences in the body from an early '31 to a late '31 car. This may be another instance of those minor differences. You might research the article and contact the author.

The '29/'30 model deck lids did overlap. If you have a possibly assembled car, it is hard saying if the parts are completely correct for your particular year. By assembled I mean that someone collected parts to "assemble" that car. Someone could have replaced the deck lid with another one that was in better shape, or that could have been a body shop repair if it was damaged. You don't show any of the structure of the deck lid. Is it entirely made of wood, or is it stamped metal that has structured stiffening raised holes in it. A '31 deck lid is metal whereas the '30 was wood with a nailing strip.  

The early '31 models did have trunk lid that overlapped the quarter panels. The parts between the lid and back window also differed. They had upholstery on the small windows instead of the metal.