Daf Yomi Moed Katan 2a: Is it Bet haShalchin or Bet haShalhin?

It is the name of the perek, after all. Almost everybody has the text with a ח rather than a ה. Thus, our Mishna and the subsequent discussion in our gemara. Thus, the Mishna states משקין בית השלחין במועד ובשביעית and the gemara states ומאי משמע דהאי בית השלחין לישנא דצחותא היא דכתיב (דברים כה) ואתה עיף ויגע ומתרגמינן ואת משלהי. So too the Mishna in Yerushalmi, and the Talmud Yerushalmi - e.g. א כל שהוא פסידה והולכת זו היא בית השלחין. So too Tosefta. So too Rashi and Rabbenu Chananel.

Still, if we look at the prooftext in the gemara, it is from Tg. Onkelos and has משלהי, with a heh. As (e.g.) Rashi points out, this is acceptable, for heh and chet switch off.

Particularly amusing in this regard is Rodkinson's Talmud online, which states

(The letter "h" is changeable for "'h"; and "mshalhi" is equivalent to "mshal'hi.")

Presumably two of those half of the occurrence of h had a dot underneath it, which did not not translate to electronic form, thus further proving the statement.
{Update: As Steg points out in the comments, chet is actually being encoded as single quote h. Thus, they are distinct.}

Why mention all this? Because Rif appears unique in all this, having a heh rather than chet in bet hashalhin, both in the Mishna and gemara. The Mishna:
כדתנן משקין בית השלהין במועד ובשביעית
and the gemara:

מאי משמע דהאי בית השלהין לישנא דצחותא הוא
כדכתיב עיף ויגע ומתרגמין משלהי ולאי

For Rif, the prooftext of the gemara needs no additional explanation. Later on, though, the word appears with a chet in Rif's citation of gemara.

Another interesting data point is Nimukei Yosef (sometimes mislabeled as Ran) in commenting on the Rif. His ד"ה has shalhin rather than shalchin, and he cites the derivation of the gemara. But he also cites a pasuk in Shir haShirim to show the word with a chet! {Shir haShirim 4:13}:

יג שְׁלָחַיִךְ פַּרְדֵּס רִמּוֹנִים, עִם פְּרִי מְגָדִים: כְּפָרִים, עִם-נְרָדִים.

13 Thy shoots are a park of pomegranates, with precious fruits; henna with spikenard plants,

then mentioning the heh and chet both being gutturals and thus switching off. Obviously, he is translating this verse differently than the English translation offered above.

Update: As S. notes in the comments, heh and chet are not just both gutturals but also share similar orthography, such that it is easy to account for a switch of one to another in the text. I would agree that this is the most likely path to arrive at Rif's girsa, or at our girsa from Rif's. Within the gemara itself, it shalchin and meshalhi are present, then the idea is that (at least at the time of the proof) the word with either letter had the same (or similar) meaning. This could arise due to either orthographic or phonological similarity, where I would lean towards the latter.