minhag - Does ב"ה actually stand for Boruch Hashem?

We asked here about writing either ב"ה or בס"ד on the top of a letter. In the question it was assumed that ב"ה stands for Boruch Hashem.

Do we have a source that ב"ה actually stands for Boruch Hashem?

That's what I was taught in school, but it always seemed strange to me that we write B'siyata D'shamya (with the help of heaven) in Aramaic, and Boruch Hashem (bless G-d) in Hebrew. It made more sense to me that it stands for B'Ezrat Hashem (with the help of G-d), as CBO1 answered, since that way the Hebrew and Aramaic match up.

Also, I've seen English letters that start "By the Grace of G-d" (such as this one), which also matches the theme of B'Siyata D'Shmaya and B'Ezrat Hashem.

[I guess you could argue that Grace can mean "blessing", such as in Grace After Meals. But that would still be closer to B'Ezrat Hashem than to Boruch Hashem, since you would be saying "By the Grace of G-d" means "with G-d's Blessing", as opposed to Boruch Hashem, which means "bless G-d".

It makes more straightforward to translate Grace in this context as "favor, or good will", than to translate it as "blessing"]

According to Wikipedia, The book "Toldot Yitzhak" (‘The Offspring of Isaac’), by Yitzhak Karo, Yosef Karo's uncle, offers the meaning of this custom of writing ב"ה (B"H), at the top of every letter, with accordance to the biblical verse: "In all thy ways acknowledge Him, and He will direct thy paths" (Book of Proverbs 3:6)

This seems to be a proof that ב"ה stands for Boruch Hashem, since we acknowledge G-d by thanking him, but I could argue that recognizing that everything comes with G-d's help is also acknowledging Him.

Also, R' Yitzchok Karo may very well be saying that ב"ה stands for בשם ה׳. See here and Alex's note below. Also, Alex points out here that the Rambam would also start sections of his works with בשם ה' א-ל עולם

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Update:

I recently came across a letter from the Lubavitcher Rebbe to the Rogatchover Gaon, that may be addressing this issue.

In the letter, Rebbe is asking for a source for putting ב"ה and בעז"ה in a letter, especially since it seems that the Gemara may indicate the opposite. However, it is unclear to me if the Rebbe is asking about putting it in the body of the letter or at the top of the letter.

So it is possible that the Rebbe was looking for a source for this as well. Also the Rebbe writes ב"ה and בעז"ה separately, so this might be a proof that ב"ה actually stands for Boruch Hashem.