Dakazman: You are correct that your top photo is factory commemorative number badge, bottom badge is not. Look at JPS # assignment list. Notice JPS numbers vs. chassis numbers are pretty random, with some early JPS #'s on late chassis #'s and vice versa. Mine is one - a late JPS #180 on a Nov 1972 chassis #2835R. And notice the random mix or R and P chassis assignments. Norwich sent Lotus USA a bag of numbers, where Europas sold very well (relative to production), and apply to black w/gold pinstripe Europa Specials.
Some were reported added to cars before shipped out to dealers (unconfirmed), while other were done at dealerships (absolutely confirmed). The result is a very random assignment. I can also confirm not every last number was ever put on any car. With that final limit of #200, not every black w/gold Europa Special got one, as we all well know.
Yes, they are treated as a sort of cool holy grail of black Specials, if you find or have one. But the truth is, the factory had no exact idea of which cars got what JPS #'s in the US. Anything now known for sure about the #'s is post facto, collected from owners who posted info somewhere, as is the case with my car. Nice to have, but really in the end, a round piece of very misunderstood plastic stuck somewhere in the interior (usually as your picture show, but mine is elsewhere).
A much as I was determined to own one and bought it almost 50 years ago, I now see it as rather misunderstood and, consequently, somewhat overrated. More or less like being every 11th caller into a radio station for free concert tickets than any actual precisely planned factory designation. Best if you were maybe on the favored customer list at a dealer, and got one added to your car as a thank you gift - on or, for some, done well after the sale.
Not that it isn't cool, but beyond a number disk, it doesn't add another feature to the car. But learning all this decades ago, I decided that keeping the car 100% original just because of a badge was not in the cards. It still has the badge, but the car has undergone multiple improvements over the years that I have never thought twice about doing to preserve a "JPS".