It seems to me that there is an argument, based on the principle that when one drinks four cups at once, he has not fulfilled the obligation and must go back and drink them in the proper order, as mentioned in the eighth paragraph, that it is not considered adding to the cups, since the four cups were not consumed according to the sages' institution. Similarly, here, since the drinking was done without the reason, the mitzvah of the sages is not fulfilled, and when one goes back and drinks with the reason, it is not considered adding to the cups.

Now, let's turn to the words of the commentary, where it is written: "And some say that nowadays it is not customary to lean, etc." I see some proof for his words from the case of a woman, who does not require the reason, as explained in the questions of Rav Achai (Sha'alta Az). This is because it is not their custom to lean, and therefore, since nowadays it is not customary to lean, everyone is subject to the laws of women.

However, this is still just a personal opinion, as written by the Beit Yosef (in the name of his father), and there is no need to be concerned about that opinion at all. As for the statement "And it seems to me, etc.," there is no need to go back and drink with the reason, as it has already been clarified that there is no concern here about adding to the cups.