after looking at the auction, i found it ended, and there was 1 offer made in the history on the bidding. the base, and xr matrix are running the 1zz-fe, while the xrs is the 2zz-ge. they are both the same size motors, being 1.8 liter, but the exact displacement is different, with the 2zz being slightly larger (1794 cc vs about 1768 cc). any turbo would theoretically work on any motor so long as you have the space under the hood, the exhaust manifold connection, the intake manifold connection, and the management (aka EMS, chip, computer, controller...and a few other names). the intercooler is not necessary, however it is very highly recommended as it does produce for better effects from the turbo.
the turbo in that auction could have been used on your 2003 xr, however you would need a different connection for the exhaust manifold. i would suggest you not get it from that person on ebay. if you are going to drop that much money, then i suggest looking for a seller who has a bit more of a reputation so you dont get completely screwed over. as a general rule of thumb, look for the rating of the seller to be equal to or greater than the dollar amount you are going to spend. once a person gets up to 1000 and has a positive rating of 98% or higher, you are pretty secure in your bid. that is just my two cents, but good luck to you in finding a good turbo kit for your car. if you have the automatic, dont build it up to more than about 200 horsepower, as the tranny in those will fall out of the ground.