I suggest a simpler emendation: "The third party accepts upon himself to acquire the goods via his own Kinyan Chatzer on behalf of the buyer," although there is not a major difference between your correction and mine. The problem here is that there was clearly a slip of the pen when we wrote "third party" the second time. The second time it should read "buyer," which we both agree on.

The other point (which possibly you hinted at when you wrote that it is inaccurate) is that Tosfos disagrees with Rashi but does not write explicitly what "Ad she'Yekabel Alav" actually means. But I think the solution is to be found in what the Ayeles ha'Shachar (that we cited immediately afterwards) writes, that according to Tosfos (who disagree with the Rashbam and maintains that one cannot transfer one's ownership of the Chatzer without an act of Zechiyah and Kinyan), the words "Ad she'Yekabel Alav" are explained to mean that the third party is Zocheh, with his Chatzer, the fruit for the second party (the buyer).

This is no doubt why you wisely wrote that "Tosfos understands," since Tosfos does not write this explicitly but this is how one has to learn in Tosfos.

Yasher Ko'ach for your comments!

Dovid Bloom