Your standard mercury vapour lamp has a starting electrode, which starts it when cold ( or when it has cooled down enough) by simple breakdown of the gap between the pointy end of this wire and the main electrode mass. The reason they do not restrike hot is the gas pressure inside is too high, after it has stopped running and keeping an ionisation path open, for this to arc over.
A metal halide lamp is just a mercury vapour lamp quartz glass body and tungsten electrodes with added metal salts and a low pressure Xenon neon argon gas fill, which needs around 2kV when cold to strike. When hot it might need 30kV to do so so will have to cool down enough. High pressure sodium lamps the same, just they use a small droplet of sodium metal ( often an amalgam of sodium, potassium and mercury) in the gas fill, though the glass used has to be a borosilicate glass so it does not get chemically attacked by the hot sodium vapour.
The size of the active cell is almost constant, it is almost the same in size from a 35W to a 250W lamp, just the electrodes and cooling for the glass and electrodes differ, and the hotter it runs the shorter the life, though it needs to run hot enough to turn the metal halides or metal into vapour with enough pressure to sustain the arc.
Xenon short arc lamps can be hot restruck almost immediately, because the construction of the lamp is such that you can apply up to 100kV across the electrodes to restrike the lamp without it arcing over externally to the lamp itself, simply because the electrodes are on either side of the body, and the lamp cooling is enough that the envelope will not melt in operation ( though it probably will be at red heat right before that), though there you also have a short life of the lamp before you replace it, as the failure mode is the internal pressure building up till the housing detonates. 200 to 1000 hours before you replace the $10k lamp is not unusual, and it will be run in a blast proofed housing, and supplied in one as well.