You're right, MS stands for manuscript, and the final M stands for Munich. It is the oldest complete manuscript of the Talmud available, which means many consider it to be the most accurate. However as Shalom pointed out, Soncino is NOT citing Jesus as being Balaam, it is citing that Jesus is in place of "sinners of Israel." I think you are seeing Balaam in close proximity to a section of censored Talmud and putting Jesus in the wrong spot. The reason that Geiger and scholars after him think Balaam is a code word for Jesus is the following:
Since the time of Geiger (JZWL, 6 (1868), 31–37) various scholars have tried to view the name Balaam, occurring in many aggadot, as a pseudonym for Jesus. They find their proof in the passage: "A certain sectarian said to Ḥanina 'Have you heard how old Balaam was?' He replied 'It is not actually stated, but since it is written "Bloody and deceitful men shall not live out half their days" [Ps. 55:24] he must have been 33 or 34'. He rejoined 'You have spoken correctly; I personally have seen Balaam's Chronicle, in which it is stated, "Balaam the lame was 33 years of age when Phinehas the robber killed him" [Sanh. 106b].'" On the basis that Jesus lived about 33 years and is called a sectarian, it was maintained that Balaam's Chronicle is none other than the Gospels and "Phinehas the robber" Pontius Pilate. However, it is impossible to imagine that a Christian would ask a Jew how old Jesus was, and call the Gospel Balaam's Chronicle or that Pontius Pilate, who is not mentioned even once in the whole of rabbinic literature, should be referred to as Phinehas the robber.
Geiger's and other more recent scholarly arguments citing Balaam as Jesus doesn't really have any concrete evidence, their argument is based on drawing connections based on similarities and oddities. However, when we are dealing with the reality of nearly all surviving texts being censored, it's impossible to find out where the censorship begins and ends. Even our oldest "uncensored" books could have already been censored, with more recent manuscripts being even further censored! Unfortunately we don't have a "dead sea scrolls" of Talmuds. So while Geiger's argument isn't definitive, those who oppose him don't have a concrete answer either because sometimes we have a clear reference to Jesus that is often edited out of most Talmuds. Such as in Sanhedrin 43, giving fuel to Geiger's argument:
‘On the Eve of Passover they hung
Jesus of Nazareth for sorcery and leading Israel astray.’ This paper
argues that other words were added to this tradition in order to
overcome three difficulties: a trial date during a festival; the unbiblical
method of execution; and the charge of ‘sorcery’.
The name Jesus the Nazerene is clearly mentioned. Sometimes when people talk about "Christian censors" what they really mean is that things were censored for fear of Christian violence toward Jews. So it could be possible that Jews themselves censored their own documents to prevent programs or other riots of violence. And when these things happen, its hard to know where the censorship began and where it ended.
If you want to know how prevalent Jesus shows up in this particular section of Talmud by cross referencing other Talmudic manuscripts, check out this website, which allows you to search by sections and compare manuscripts, I suggest you look up the section of Gittin you're curious about: http://jnul.huji.ac.il/dl/talmud/bavly/selectbavly.asp
Here is a photo of Gittin 57 in question in the Munich Manuscript. You can clearly see the name of Jesus if you look three lines above the large word אתרנגול
It states that Jesus was raised from the dead through incantations: אסקי' לישו בנגיד'. In the standard text this has been altered to read לפושעי ישראל, instead of Jesus.
Sorry that the quality isn't the greatest. If you want to get a more readable version you can directly view the Munich manuscript here. Unfortunately it's not broken off into daf numbers.
When it comes to these arguments there will never be a definitive answer. Because it comes down to things like this: If I am an "Orthodox Jew" and have a belief that only x manuscript of the Talmud Bavli is the correct one, then i have no concern if there are other manuscripts out there that explicitly mention the name of Jesus.
Another issue comes down to this: We have hundreds, sometimes thousands of manuscripts, and most of them vary, so how do you find out which is the correct one. You could make the argument that when most of them agree on something, then the minority of manuscripts that differ are the variant. But when the question comes to the potentiality that something was censored, how do you know which is the actual original, the one that has the rare reading, or the ones that all conform to something? What if the oldest manuscript we have (the Munich) has Jesus in several places, whereas every newer manuscript has the name Jesus missing? What do you do when most of the Ashkenazi manuscripts say x, and the Sepharadi or Yemenite ones say y?
I'm interested if anybody before Geiger claimed that Balaam was Jesus. How far back the claim goes and who made it.
Undoubtedly when you have such similarities in the texts about Jesus and Balaam there will be lots of people who make this argument. But more than likely most people who made this claim in older times probably weren't Jews, as Jews try to talk as little about Jesus as possible. This can lead Rabbis to ignore sections of any manuscripts that could be too problematic.
For more information regarding Jesus in the Munich Manuscript I recommend that you read this article: Jesus of Nazareth's Trial in the Uncensored Talmud. Written by a prominent scholar in Rabbinic Works.