[Avodah] What is the source for the minhag of Chasidim to have HaKafos on Shmini Atzeres night?

[Simon Montagu]

On 10/24/06, Micha Berger <micha at aishdas.org> wrote:
> For example (learned from RGDubin back in v4, 2000ce), the Torah Temima's
> (since disproven) theory about the origin of saying "migdol" in bentching on
> Shabbos. He writes that the original note was to contrast the text of
> bentching with that in Shemu'el beis, and therefore read "besh"b 'migdol'".
> Somone then expanded the rashei teivos to "beShabbos" and so the minhag was
> allegedly born.
>
> (WADR to the TT, the theory doesn't work because the Avudraham mentions the
> practice, and he didn't know about a future splitting of Shemu'el by people in
> another part of the world.)

I don't understand this disproof. The TT's theory seems to me to work
just as well if the original note said "beit shin" for "bishmuel"

A similar example that I heard from the late Rabbi Ephraim Wiesenberg
of London is the note in some haggadot that on motza'ei shabbat one
should say "min hapesahhim umin hazebahhim" instead of "min hazebahhim
umin hapesahhim" shortly before the second cup of wine. Now, there is
an explanation (which I forget) why on motza"sh the pesahh was eaten
before the zebahh, but what does that have to do with it? In context,
it refers to "regalim ahherim haba'im likratenu" and davka not this
year, so why should the order this year be taken into account?
Instead, it is suggested, there was originally a note "beit mem shin"
meaning "bamishna" (referring to the mishna within the bavli, Pesahhim
116b) which was misinterpreted as "bemotza'ei shabbat".