Is a yeast starter made using only the wort for the batch being brewed?
No.
........swirl the bottle to suspend everything and pitch the whole thing.
I never thought about it, but "starter" is not quite the right word.....perhaps "multiplier" would fit better. As Preston described, you're trying more to multiply your yeast cell count in advance of brew day to fit the style and batch size you are brewing. Higher OG's and higher volumes of wort demand more yeast to complete the fermentation with the yeast character you seek (clean versus fruity, spicy, phenolic, etc.).
Some people pitch the whole thing, but others do it days in advance and chill the starter liquid to settle the yeast so that they can decant the spent wort. With a big starter, especially in a subtle beer like kolsch, the spent wort in the starter could throw off your beer.
www.mrmalty.com has pages on starter methods that are worth reading.