Color temperature is measured in degrees Kelvin (K) - temperature relates to color through the light emitted by a black body at a given temperature. Things need to get quite hot before bodies emit enough visible light for us to notice. For example the bars of a electric fire will appear red or orange when hot. If you kept raising the temperature of the fire, it would appear white then blue, if it didn't melt first. I don't think hot bodies ever look green, although at some point green must be the peak color being emitted, at this temperature things look white because of the balance of red and blue light.
seagull and rainbow
The light used for photography is being emitted by hot bodies of some kind. Either the sun, a flashgun, fire or lightbulbs. The situation is complicated because light gets filtered or reflected before it reaches the subject. A camera sets the color temperature to about 5000K for a low sun, and 6000K for sun overhead. But it's the same black body so the color temperature is just being used to approximate the effect of the atmosphere in absorbing light. Similarly light in a forest floor is fitered though green leaves, what the camera does exactly with AWB I don't know, but I don't think any color temperature is going to get rid of the green cast properly. Then you can have a mixture of natural and artificial light in one scene, which will inevitably confuse the AWB.
So in practice, color temperature is the first step in obtaining true colors, but there are situations where more color adjustment is needed. But I often find that if I can't get good colors from any color temperature setting, then no amount of curves adjustment can make things look natural. So it's important to tweak the WB in the RAW conversion to get the best result possible.
What people call warm colors - red, orange, and yellow - correspond to low color temperature. And cool colors - white and blue - correspond to high color temperature. I think this has happened because we rarely see things get white-hot or blue-hot. Red-hot is about as far as it gets. Blue looks cold because it is associated with twilight and perhaps with ice. But there's no doubt the color balance changes the way people feel about an image. The shot below is taken with a telephoto lens looking into the sunset. The AWB gives me 3150K and makes the shot look grey, which might be technically correct but does not convey the atmoshpere I want, I have used 5500K instead.
sunset over Lake Thun, Switzerland
The picture below was taken on an icy day in Canada. That was back in days when I used a D30 and shot everything in jpeg. So unfortunately I can't tweak the color temperature on this. But an auto-levels adjustment has given me blues in the background. I like to get some blue tone in a winter scene but not necessarily a blue cast.
snow on teagles