"Much to my delight, I found I could interact with a compass to find my bearings, view images, and use an on-screen teleprompter to read a surprising but familiar quote."

It would seem difficult to view images or read a teleprompter via 4 pixels. More likely they mean exactly what they wrote - 0.02 inches at 14,000 pixels per inch.

Did I math wrong?

    0.02 in * 0.02 in == 0.0004 sq in
    0.0004 sq in * 14000 ppi == 5.6 pixels

It's probably 14000 ppi, not pixels per _square_ inch. So that's 0.02 in * 14000 in each direction (which is 280 x 280 pixels).

You assumed square. Round gives 4 pixels:

π(.02/2)^2 * 14000 = 4.4