These "Black&White" pictures contain black and white pixels in a checkerboard pattern. "White&Black" pictures are similar, only with swapped colors.
When you view them zoomed-out, filters in your browser (or in whatever image viewing program) may create gray color and/or moiré pattern.
When you view them zoomed-in, blur may or may not be applied, it introduces gray. Moiré is also possible, depending on filters.
When you view them with 100% zoom, i.e. when there is a strict correspondence between image pixels and LCD pixels, each LCD pixel is supposed to be either black or white. You may still perceive the image as uniform gray color because LCD pixels are very small (compare: dither).
everytime I slightly move the window and leave it a certain spot on my screen, the pixel comes back to life! however, when I move the window again, the pixel "dies" and goes back to being red...
This is expected, especially with 100% zoom. When each pixel of the image is either black or white, the dead LCD pixel is supposed to be either black or white. And you said:
every time I display a blue or green background, the pixel is completely dark. The pixel is red when I display a white background.
So the pixel can be black; but if it's supposed to be white then it will be red. With a checkerboard pattern of black and white you can get black or red from the pixel, depending on what color the pixel is supposed to be. By moving the picture you changed the color the pixel was supposed to be. When it was black, it blended in the pattern and created "uniform" (dithered) gray. When it was supposed to be white, it stood out as red.
To be clear: the pixel did not revive, it did not die just after; it only changed colors in scope of its limited abilities.
Note such checkerboard patterns, when it comes to dead pixels, are meant to be displayed with 100% zoom and not meant to be used as static pictures.
In case of diagnostics you need dead pixels to stand out. You learned that while displaying a static checkerboard pattern a dead pixel may or may not stand out. That's why you need to move the pattern around or cycle between patterns.
In case of a repair attempt, you base on a belief (true sometimes) that a stuck/dead pixel can be revived after changing its desired color quickly. Naturally you cannot do this with a static image. Programs that may help generate non-static images: they toggle colors, cycle through colors or provide noise.