| OK, this house is red. OK, it's the third house if you move from the east (but the directions said move from west to east, not east to west). From the west, this is the third house that is red, not the third house, which is red (or happens to be red).
When we say the “third house that is red,” it means you're looking for the third red house, meaning some other-colored houses might be in the mix before you get to the third one that is red. One red house, one white house, a second red house and, finally, a third red house. When you use “that,” you must know the color to identify the house.
But you are looking for the third house, which is red — which means it just happens to be red. You don't need its color to identify it.
It's a big difference — which means (shame,shame!) you've just killed the the Cleaver family: Ward Cleaver, his wife, June, big brother Wally and, of course, the Beaver. Oh, that damned Eddie Haskell was there, too, but that part is OK because nobody liked him. (And does anyone know what I'm referring to? It's “Leave It to Beaver.” Think the Simpsons in the '50s without the sarcasm or social commentary.)