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I can't answer the question other than by observation... knowing that the "sodium" bulbs are lower on the color spectrum and appear more orange as compared to the quartz halogen lamps or metal halide bulbs.
I can however speak with regard to color temperature (rather important from a color rendering point of view).
Color temperature values are based on a standard where 5,000 Kelvin is basically the temperture of natural sunlight at the base of the Washington Monument at 12:00 Noon on July 4th. (No kidding!) Don't know when and how they arrived at that, but surely it was on a clear day with no smog.
So, values on lighting, in addition to lumens (brightness), is often provided with color temperature values in Kelvin. The lower the temperature, the more red/orange and the higher the temperature, the more blue.
Quartz Halogen lamps have a temperature in the range of 3,600 Kelvin and "Sodium Bulbs", I believe, are less than 1,800 K. Quartz "Flash" tubes which are used in photography are color corrected to be as close to "daylight" (5,000 K) as possible. While standard florescents are available in many "colors", you can buy color corrected flourescents that are close to "daylight". Tungsten filament bulbs (incandescents), as I recall are around 2,000 - 2,400 K, but I probably should look that up. Most all bulbs (Tungsten, quartz, florescent, etc.) change color over time, usually going toward the red/orange as they age. If you need to see colors correctly, that's important, bu if you're simply lighting up the yard, it probably doesn't matter much.
The bottom line is really that brightness (lumens) and color value (light temperature in Kelvin) are different values. Generally speaking, a very bright (higher lumens) bulb with a lower temperature will not appear as bright as a higher temperature bulb with the same lumen value. I am under the impression however, that lower temperature bulbs require less power, and last longer, hence the growing trend toward high pressure sodium in commercial lighting such as warehouses, parking lots, and street lighting. But, in commercial retail applications, where customers need to see better color values, lighting with a higher color temperature is a must.
A bit long and probably more than you wanted to know,
I'm not familiar enough with Metal Halide or High Pressure Sodium lighting to answer that question. My only familiarity with the MH lights is that they are popular for use with aquariums because they enhance the color of the tropical fish. With HPS, the company I retired from, used them in our shops and the machinists hated them. They saved the factory a bundle because of their lower wattage, but the orange light seems to dull everything, as compared to the older, much whiter, but grossly more expensive lighting system.
So, I may be wrong but I don't recall any significant warming up period with the new HPS shop lights. With quartz halogen lamps, they're instant. (They're used in most every automobile headlight. They do produce a substantial amount of heat though (Quartz Halogen) and you have to be careful when changing bulbs that you do not touch the glass with your bare fingers. (Touching the glass may contaminate the glass with skin oils, etc., and because of the extremely high heat, the glass will heat unevenly and could very well shatter.