So let me get this straight. Cloud-filtered sunlight and shade are cooler than standard strobes, so you want to cool your strobe to match the ambient. Similarly, sunlight and evening light are both warmer, so you want to warm up your strobe light to match that of the ambient.
Exactly. Or as I like to put it, "Make sure the light coming out of your flash is f-ed up in the same way as the ambient light." Then, when you make corrections in post, both the ambient lit parts of the image and the flash-lit parts correct equally, and the result should look fairly natural.
I work fairly frequently in rooms lit with tungsten light, about 3200K. The full-CTO is a frequent flyer under those conditions. Also remember that balanced colors are only half of the equation. The other half is to soften or diffuse the light and have it coming from a believable direction. Somewhere off camera is usually much better, depending on your situation and the look you want.