Volumes have been written on the color of light. Scientists have been studying it since ancient times.
Ultimately, a person's choice of color is his personal preference. Some people like warmer tones, some like cooler tones. In practice, most people would choose a warmer tone for their living areas to create a comfortable ambiance. Most would choose a cooler tone for their garage or workspace. But either way, it's just preference.
There are a few specific reasons to use one color over another. An example would be medical work, where it's important to be able to distinguish blue veins from red arteries.
But other than those few specifics, it's just a preference.
Some folks think that there is more light being delivered from the cooler color lamps, like 5000k. Normally, this isn't true. It's just that your eyes respond better under certain conditions. And everyone's eyes arent equal, in fact, as you age, your eyes change. So your preference could change, too.
The color of delivered sunlight changes dramatically throughout the day depending on weather, time of day, latitude and longitude, and pollution. There is no one color for delivered sunlight. For consistency, some industries, like the film industry, have chosen a particular color to be the standard.
The lighting industry has no standards for the lingo they use to describe the color of a lamp. Words like soft white, cool white, daylight, natural white, etc...mean nothing. As the OP pointed out, the color temperature in kelvins is what you need to look for.
The picture is a good representation of the difference when used in a garage.
Interestingly, color rendering (CRI) is very different than color temperature (CCT). A lighting expert wouldn't necessarily look at the color of the lamps, but they would look at the items that are being lit to see how well they are rendered compared to natural light. To me, this is a more important criteria.
OP- Good pic. Thanks
CD