Generally a cool light is better for task lighting in the kitchen but you might find the 6300-7000k a bit harsh. I personally recommend a light that is more natural (about 4000k) for kitchen. You should also keep in mind that if your kitchen is open-plan with your living space it is better not to mix warm and cool lighting. Hope that helps.

When selecting the ideal color temperature for lighting, our Design Team typically starts at both the style and decor of the room. Homes with a more traditional style, featuring dark woods and earth tones tend to look best with the warmer white (anywhere around 3000K), while more modern spaces tend to look better with the more fluorescent looking cool white light (closer to 6000K). Contemporary kitchens, or those with a lot of white can benefit from a more neutral white (around 4000K).

That said, it is also important to keep color temperatures consistent throughout your home. If your kitchen is already fitted with traditional incandescent bulbs, the warm white will likely match better, while cool white will look better alongside fluorescent bulbs.

Below is a graphic we use to help our customers better visualize their lighting options. Hope this helps!

There are no rules. Use what ever color temp floats your boat. Color rendering is more important than temp. Different colors in different rooms or at different levels in the same room (floor and table lamps warmer than ceiling is what I like.) are fine. That page that you reference gives no color rendering quality information so it probably sucks.