What I’ve noticed with HPS lighting, is that some bulbs start up differently. Some bulbs will start a xenon color and then turn a bluish white color while warming up (and they don’t turn yellow-orange until they’re full brightness), while others will start a xenon color and then shift to a deeper orange (like an LPS), and then turn a normal HPS color.

Some of them will even start a xenon color, shift to a bluish color, shift to orange, shift back to a bluish color, and then shift back to the orange color.

This is probably related to the sodium:mercury ratio in the sodium amalgam. As the lamp ages, there is more mercury in the sodium amalgam, so the lamp starts bluer.
Also: There are unsaturated HPS lamps that starts mercury color, as well as mercury free HPS lamps, which starts orange.

Yes absolutely! I have a heavily aged, near-EOL 250W SON-T, and during power up it starts off as a 5000K-white, almost MV looking arc with a subtle green tint. Eventually the sodium kicks in and the rest follows. However my brand new 250W SON-E starts off as a near-monochrome yellow and never really goes through a white phase. There's one pole on my street that starts up as what looks like a heavily HPMV arc, and the arc tube is heavily blackened/near-EOL as well.

There are quite a lot of differences among various HPS designs even when of the identical rating. Differences in geometry, temperature balance, exact designed operating temperatures and pressures, fill design (amount of Hg, buffer gas pressure and composition), the way how exactly is the operation stabilized (unsaturated vs saturated designs, the way how the pressure is regulated with the saturated types) and so on, each then impacting the light output, mainly when not stabilized (warmup,...) in some way or other.