If you don't know what I mean, I mean that from the moment you turn it on until it hits the orange color. Basically, that small portion that has that cool white color. Any ideas?

I’ve found they only tend to stay white longer as they age?
When new all sizes to me have the same ‘white’ time when running up.

I see, but I'm also asking this because I've heard some brands of HPS lamp have slow startup sequences. I had made a comment on a street light video like two years ago with a question that was similar to this topic and someone mentioned a brand but I don't remember now.

Consider that with the push to lower the Mercury content, the lamps do contain less Mercury..

The absolute longest i seen were on the road up from my house. Those were some 250W SON-T from unknown manufacturer, original first lamps installed in Hubbell RMG lanterns with 230V 50Hz gear, from the late 80s

Most of those lamps lasted well into the 10's (everything was replaced with LEDs in the 10's)

Some lamps would warm up to fairly bright clear Mercury light and stay at it for a good 10 minutes or more before continuing to Sodium color. One of them was occasionally cycling in the later years, it could cycle one night and remain stable for many more nights in a row. The off periods were very long (to the extent unusual even for MH), the restrike was directly from nothing at all to the clear Mercury color, no glow and often no flashes before the restrike

I’ve found they only tend to stay white longer as they age?
When new all sizes to me have the same ‘white’ time when running up.

That is correct.  Especially for high pressure sodium. 
Some brands will start up and warm up quicker as well.  Sylvania would warm up fairly quickly versus GE which wasn't bad.  Philips seemed to take a little longer.