An LCD/computer screen already emits linearly polarized light, so a linear polarizer in front of it will typically behave like case 1: as you rotate it, the screen goes from bright to nearly black.

A circular polarizer is different: it is a linear polarizer plus a quarter-wave retarder plate. If the screen’s light hits the linear-polarizer side first, it behaves much like a normal linear polarizer and can go dark. If the screen’s light hits the retarder side first, the result may not fully extinguish and can show color shifts instead.

So the black-vs-color behavior is not simply “linear vs circular” by name alone; with a circular polarizer it also depends on which side faces the screen. The color shift happens because the retarder plate is not perfectly neutral at all wavelengths, so different colors are affected slightly differently.

This is also why circular polarizers are used on many cameras: they avoid problems that plain linear polarizers can cause with some autofocus/exposure systems.