when I specify the image size in pixels to start with, it seems to me that any figure entered in that ppi box has no bearing whatsoever on the file I export or how it displays or prints.
For me, working mostly for offset print, an image usually ends up in an InDesign file. There the size is set and an
effective ppi calculated from the pixels in the file. What the
actual ppi is, as set in the file metadata, is completely irrelevant.
Similarly, in a printer driver (and Lr print module) the image is usually scaled according to paper size, so the ppi setting is simply ignored. Or so it seems to me (I rarely print myself).
In principle, it's a simple equation: resolution = pixels / size. And like any equation, it can be inverted and turned upside down, so that you can start with any two values and calculate the third. I just can't think of any situation where it actually matters...
For screen it's moot. There you already have a pixel grid, the screen pixels, so it's just a matter of mapping image pixels to screen pixels.
EDIT: I once got a request for banner size "at 300 dpi" (sic). For fun I calculated the pixel count for such a monster, and it clocked in at some 35 000 pixels long edge. I still had a hard time convincing them that 300 ppi wasn't
really what they needed...