Have you heard this one? Paddy Englishman, Paddy Irishman & Paddy Scotsman walk into a bar........................................
Seriously I've never seen the term "shadow" used in the context of "screendoor" (no doubt you'll now post several links to prove me wrong).
There are no "shadows" on my (totally flat) screen caused by the PJ. The dark grid visible on screen is caused by the "gaps" between light emitting pixels (well they don't actually output light - the bulb does, they act like filters changing the incoming "white" light to red green & blue - 3 LCD panels).
The bigger the gap between pixels, the more noticeable is the grid on screen (well - for some). Things such as defocusing & fitting filters (Hoya FL-Day) can help to aleviate the effect.
TBH, it is very subjective. A demo IS advised for anyone who doesn't understand what "screendoor" actually looks like.
While your at it you may want to look into LCD/Transmissive electrode technology