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Yes the different versions show lechaora a machloket regarding the definition of Bitul Isur Lechatechilla.

טיפת חלב שנפלה על החתיכה, אם יש בה בנותן טעם באותה חתיכה, אסור. ניער את הקדרה, אם יש בה בנותן טעם באותה קדרה, אסור.‏

A drop of milk, that fell on a piece of meat, if its able to give taste in this piece, it's prohibited. If he shook the pot, if it's able to give taste in the pot, it's prohibited.

The version of the Mishna following the Rambam is different.

ח,ג  טיפת חלב שנפלה על חתיכה--אם יש בה בנותן טעם באותה< חתיכה, נוער את הקדירה; אם יש בה בנותן טעם באותה הקדירה, אסור.‏

A drop of milk, that fell on a piece of meat, if its able to give taste in this piece, one needs to shake the pot, if it's able to give taste in the pot, it's prohibited.

Two differences:

  1. The first statement, ruling that if the milk drop is able to give taste in the meat piece, it prohibits (or the piece only, or the whole pot's content),
  2. There is allowed to shake the pot, even if one knows that this action lead to the annulment of an isur which is already happening.

We can assume that what is recommended in the second version isn't following the supporters of the first version.

The Bartenura, following Rashi, comments on the second statement of the Mishna according to the first version.

ניער את הקדירה - שהגיס בה מיד קודם שקבלה החתיכה טעם מן הטפה, דהשתא נתערבה הטיפה בכולן:‏

Shaking the pot is useful when one shake immediately, before the drop impregnated the meat piece.

אם יש בנותן טעם באותה קדירה - כלומר אין כח בטפה לאסור כולן אא"כ יש בה טעם ליתן בכל הקדירה: ‏

It's so consistent that the Mishna says that the result of the shaking depends if the whole pot is great enough to annul the milk drop. All is allowed or all is prohibited.

The first statement of the Mishna describes what is the rule when one has not been fast enough to shake the pot immediately. Bartenura says:

כלומר אם אין באותה חתיכה לבדה ששים לבטל הטפה, מיד נאסרת החתיכה, ואוסרת חברותיה וצריך ששים כנגד כל החתיכה

If the meat piece on which fell the milk drop is not (volumetrically) sixty time bigger than the milk drop, the piece is immediately prohibited. And from this moment, if the rest of the content of the pot is not sixty time bigger than this piece, the whole pot is prohibited. The meat piece prohibits the entire pot because one shaked the pot after it absorbed the milk (Tosfot Yom Tov explains this from the Gemara) .

In summary, the Mishna enumerates two laws depending on whether the shaking was before or after the time needed for the drop to give taste in the meat piece.

But the Mishna according to the Rambam gives advice what to do to avoid to make tref the pot (Dikduke sofrim).

I am pretty sure that the version of the Rambam vehicles a "novelty" not universally accepted, regarding the topic "one does not annulate an isur Lechatechilla".

The same statement of Rambam is in Mishne Tora (Maachalot Asurot 9.10), and is quoted in Shulchan Aruch(YD 92.2).

I found it in the Sifte Cohen sk 8.

ואין להקשות היאך ינער הקדרה שיתערב הכל דלמא יש טעם בחתיכה אחת ואיך יבטל איסור לכתחלה וא"כ הוי ליה למימר שיטעום העובד כוכבים כל חתיכה וחתיכה בפני עצמו יש לומר דכיון דלא נודע לאיזה נפל הרי נכנס כל הקדרה בספק ואין כאן משום מבטל איסור וגם דמי למה שכתב מהרא"י בת"ה סימן קע"א ואין זה מבטל איסור לכתחלה כיון שספק הוא אם יתערב שום איסור כלל עכ"ל והבאתיו בסי' פ"ד ס"ק ל"ח וכן לקמן סימן קט"ז ס"ק כ"ח וה"נ ספק הוא אם יש טעם בחתיכה זו ונראה דגם הטור הבין דעת הרמב"ם כן אלא ס"ל כיון דאפשר למיקם עלה דמילתא א"כ לא נכנס הכל בספק שהרי יודע הדבר על ידי שיטעום העובד כוכבים כל חתיכה וחתיכה בפני עצמה והלכך כיון שחתיכה א' היתה מתבררת בודאי שהיא אסורה א"כ מאי מועיל ניעור אח"כ

He objects against this practice (to shake lechatechilla) that it may be a problem of annulling isur Lechatechilla (1). And answers that the whole set of pieces it in safek isur situation, perhaps the drop of milk is already annulled in the first meat piece. So this is annulment of a safek isur. And with safek isur there is no bitul isur Lechatechilla. But for the Tur and Rema, the objection is right. Indeed, one can verify each of the meat pieces (there are upper the water's level) if it have milk taste, when one can check the situation, the din is not safek. So this is mevatel isur Lechatechilla.

See in Peri Megadim who addresses the issue of bitul isur Lechatechilla on his comment on the Shach called Sifte Daat. There are a lot of details in the Sugia, e.g. the different steps of the shaking. I wanted only to point that the version of the Mishna of the Rambam is not mentioned in poskim, but seems to be a reference for the Din of Rambam and Shulchan Aruch, despite that Poskim didn't address it.


(1): To be more precise. If the drop of milk fell on a small piece of meat, there are two possibilities of problems. 1) If it didn't still be absorbed, when one shake it, we add meat to annul it. 2) If it's still absorbed, we have a prohibited piece of meat that "became nevela" as says the Gemara, and we annul it with the others pieces. But we don't know what happened. The first problem is not discussed here in poskim, may be, because there is no isur at all at this stage. The second is discussed. The teruts of the Magid Mishne is that when we cannot identify the piece, there is no din "piece became nevela". The teruts of Mahari Ben Chaviv is that because he cannot identify the piece, the pieces are as if they are mixed and one can shake them.