If CaT is the same as both CaW and CaS, as you state in the quote, that implies you see CaW and CaS as being the same.
They're not.
That said, in my mind CaT can certainly overlap both CaW and CaS (I noted this way upthread as well) and I also think CaT can be its own thing: a combat that exists in the fiction mostly so both the PCs and villains can say dramatic words, strike dramatic poses, and develop their characterizations through so doing. Hell, the actual combat in CaT doesn't even have to come to a final resolution...
Villain (winning the fight but nobody's anywhere near dead or dying yet): "I have blooded you. I have bested you. I have shown you for the pathetic creatures you are! I leave you now to wallow in the misery of your defeat! Oh, and rest assured that should we meet again the same result will follow, and worse!" (villain exits the scene by whatever available means he has)