Some HPS lamps don’t come back on until the bulb’s arc tube goes out completely or nearly completely, others will perform a hot restrike while cycling and come on nearly at full brightness, others will do that at lower brightness.

Some HPS lamps come on straight to orange, some cycle in only a blue color, some cycle in blue and shift to orange before going out. Also, some HPS bulbs flicker like failing LED bulbs when they’re failing.

There are multiple variations in the exact lamp chemistries (the amoiunt of Hg vs Hg-free, the buffer gas composition), the lamp construction (unsaturated vs saturated, the way how the temperature is controlled). related to this is the exact lamp position within the lantern and the way how the lantern optics does vs does not reflect the heat radiation back onto the arctube. Mainly with the latest even small variations within assembly tolerance can lead to significant differences in behavior.
Then there are differences in ignitor behavior: Basic ignitor can to some extend support the arc burning even when it won't hold on its own, this may lead to "brightness switching" or fast cycling (multiple per second,...), but some ignitors are equipped with various cutouts with timers, originally intended to properly restart the lamp and prevent some "half burning" after e.g. mains dips or so (then the lamp may first reduce brightness for a few seconds and then shut down, cool down and then restart), some have dedicated functionality detecting failing lamp and blocking further restarts, intended to prevent the cycling (then the faulty lamp just warm up once on power ON and then shut down).

And then there is something what could be called an "individual attitude" of the lamp. Tthese effects happen when the lamp conditions become unstable. And when unstable, even tiny differences in the exact condition or tolerances may cause significant difference in resulting behavior.