First, the instructions at the top (circled in red):
"The three 'judges' sit while the petitioner stands before them and states:"At issue is how the hatarah works. Based on their commentary, Artscroll thinks it does not work at all. In which case I suppose it makes sense to have the judges sit. The purpose is to make the person realize the severity of his sins by vowing, and the more formal we can make it, the better.
However, assuming that is actually supposed to work, there are two means an annulment can happen. One is to find a specific petach, an opening, such that if the person had known X, he never would have vowed. The other is to open the vow with charatah, regret. It seems that the idea within the formula is that they are opening with charatah. Whether the particular brand of charatah, of regretting the making of the vow but not the particular actions, is valid, I will not address here, but certainly, this seems to be the intent of the formula. The difference lehalachah between them is that for the former, real deliberation must be done, and so the judges must sit. But when we open a vow with regret, no deliberation need be done, and so the judges may stand. And yet the directions here are that the judges sit. I do not understand this requirement. The directions may be a matter of custom rather than halacha.
While on this subject, all sorts of other restrictions of a bet din are relaxed, for these are not judges but rather "judges," as the machzor makes clear. Thus, hatarah can be done at night, or on Shabbos if the neder needs be annulled for the sake of Shabbos. And the judges may be related to one another.
My next nitpick is with the formula the judges use in responding to the petitioner. According to the Rambam, the judges only need to say one time, one language of annulment. Thus, they can say muttar lach, or they can say machul lach, or they can say sharui lach.
The Tur (IIRC) has a more developed formula. Once again, they can say muttar lach (or they can say machul lach, or they can say sharui lach). But whatever they say, they say three times. Thus (see here)
טור יורה דעה סימן רכח
Or according to Shulchan Aruch:
כיצד היא ההתרה, יאמר לו ו ד] ג' פעמים: מותר לך, ה] או שרוי לך, או מחול לך, (<ד> בכל לשון שיאמר), (ב"י בשם הרמב"ם), ז ו] <ה> אפילו מעומד, (ב) ובקרובים, ובלילה, ובשבת, אפילו אם היה אפשר לו מאתמול לישאל עליו, ז] ובלבד שיהיה לצורך השבת, כגון שנדר שלא לאכול או ליבטל מעונג השבת. ח ח] <ו> וחרמי צבור, נהגו להתיר אף על פי שאינם לצורך השבת.
ג' פעמים - לאו דוקא דבחד זימנא נמי סגי אלא עושין כן כדי לחזק הענין עט"ז ומשמע דבדיעבד סגי בפעם אחת וכ"כ הב"י והב"ח בשם הרמב"ם דא"צ שיאמר רק פעם אחת:
The nusach in the machzor has all three leshonos, and the judges repeat the whole formula three times. Thus, there is a total of nine statements. Plus we have the whole paragraph, not just those three phrases, which they recite three times. This is exceptional overkill. But then, I suppose it makes for great ritual. Ritual likes repetition of threes. As you might guess, I take exception to this. (Perhaps we might say this is kedei lechazek hainyan to the nth degree. Or rather the threefold repetition is kedei lechazeik hainyan while the use of all three languages, plus more, is just the way that liturgy grows way past what is required...)
While on this subject, I might note that again lehalacha, it seems that the bet din can annul vows of multiple people, just as they can annul several vows simultaneously (though there is dispute about the latter, about several vows where one is chal on top of the other, such that it only takes effect once the first one is annulled). So the typical process in shul is possibly a tremendous time waster. It depends. If people always pair up in threes, then they get out pretty quickly. But an acceptable alternative is to let several people say their petitions, and have the bet din then grant the annulment to all of them.
Finally, a word as to the translation of hakol yihtu muttarim lach, hakol mechulim lach, hakol sheruyim lach. Artscroll translates
"May everything be permitted you, may everything be forgiven you, may everything be allowed you."I forget where I saw it, but I seem to recall other explanations. Mutar as in untying, as with a knot, and the same for sharui. And for machul, not in the same of forgiven, but rather making profane, as in lo yachel devaro, and the associated derashot.
Note: Not halacha lemaaseh. Don't pasken or act based on blogs.