Color temperatures (CCT) in product labeling: which is appropriate?

You're not alone. I noticed this too. Obviously this is not an appropriate classification of correlated color temperature (CCT). The 6000-6500 K range is a rarely used CCT in many markets such as the European countries and the United States for psychological and photobiological safety considerations.

Color temperature is designed to characterize the apparent "tint" of the illumination (e.g., warm or cool) generated by an electric light source. The color temperature of a light source is a numerical measurement of its color appearance. Color temperature is commonly described in degrees Kelvin (K), a unit of absolute temperature. Lower color temperatures generally are thought to be "warm" as they contain more yellow-red colors, while higher color temperatures generally indicate white light having a substantial blue component or a "cooler feel." By way of example, the CCT of an incandescent lamp is usually within the range of 2700 K-3300 K. Natural light sources include daylight with a relatively high color temperature (e.g., about 5000K).

The perceived feel of color groups is the psychological impact of lighting and directly affect people's choice on selecting the color temperatures of a luminaire. For people living in countries in the low latitudes where the climate is hot such as Thailand, Brazil, Kenya, and Indonesia, they may prefer lights with high color temperatures as these bring them a cool feel. Vice versa, in cold climate countries lights with low color temperatures are more popular. Neutral to cool color temperatures (4000-5000K) are generally appropriate in office space because they are refreshing and productive. Warmer color temperatures tend to work well in dining areas, guest rooms, and living rooms as they create a warm, inviting atmosphere which makes people feel relaxed. Consider lighting in the 2700-3500K range.

Color temperature has an enormous effect on our physical and mental well-being. The reason I do not recommend manufacturers to deliberately label in their product brochures the high color temperatures such as 6000-6500 K is for photobiological safety considerations. Advances in solid state lighting technology have contributed to the development of LEDs with a satisfactorily high light output, making them an appealing alternative to incandescent or fluorescent lamps. However, the "white light" in LEDs is typically produced by mixing the blue and the yellow light. Namely, a light emitting diode generates blue light and a phosphor emits yellow light in response to excitation by the blue LED output, whereby the blue and the yellow light, when appropriately mixed, produce light that is perceived as white light. To avoid the potential blue light hazard such as retinal injury, CCT over 6000 K should not be recommended for interior lighting by the manufacturers.

As to the color groups, I'd suggest they're categorized into these ranges:
Warm White (2000 - 3499 K)
Neutral White (3500 - 4099 K)
Cool White (4100 - 4999 K)
Daylight (5000 K & Above)