>I'm not sure how either term is used in the case of multiplication
(I grew up in India.) I imagine it was just "multiply 2 into 10" that got shortened into "2 into 10", and then "2 into 10" by itself became a noun that represents the product, so "What is 2 into 10? 2 into 10 is 20."
"How many times does 2 go into 20" is 20/2, which is shortened to "2 into 20".
"20 divided by 2" is also 20/2, which is shortened to "20 by 2".
I'm not sure how either term is used in the case of multiplication however but I'm pretty sure that's how the division side of it works.