Oh my goodness, that Ba'ch -- it's a solid page and then some. So the Tur holds like R"T and so paskens the Rama, that if a nonJew brings something on the first day of YT, you have wait the amount of time after the second day YT for the article to have been brought.
But: if the the first day was really YT, then soon as the second day is over it should be good enough and if the second day was really YT, then it was brought on "chol" so again soon as the second day is over is should be good enough. In other words, either way, why wait the amount of time that the article could be brought on motzei second day YT and not just soon as YT is out?

There are two reason why you have to wait the amount of time it takes to do it on Motzai Yom Tov, 1) in order that you not have enjoyment from work done on YT...According to this reason you are indeed correct and it would be permitted on the second day 2) [which is the reason of RT] that the sages were afraid that if it is permitted to have enjoyment from the work done immediately after Yom Tov from the work done for you on YT, you might tell the non-jew to do the work for you on YT in order that it be ready right after YT, according to this reason, you DO have to wait until after the second day YT enough time for the work to have been done, for we are still scared that you might tell the non-jew to do the work on the first day to have it ready on the second [or right after the second] day.


ETA: why as the word "IT" censored out on first try !!!???