I think this phrasing is less confusing when it refers to labels rather than actual numerals. For instance 'divide the price into the exchange rate', which would mean the same as 'divide the exchange rate by the price'.

Why would you use this phrasing? I suppose it can flow more easily in some cases, for instance 'first work out X, then work out Y, then divide them both into Z and see which one is larger' is less of a mouthful than the alternative.

It comes from "how many times does 2 go into 4", which is "4/2".

If you go to a pizza restaurant and ask them to "Divide two (large pizzas) into four (portions)" they would indeed give each person 1/2 pizza. So for colloquial use you are correct. Which makes the math definition confusing.

Perhaps it is some kind of regional difference. I've always lived on the US West Coast, and I've heard "divide four by two" expressed as "divide two into four" many times - and never the other way around.

I never use the "into" phrasing myself because it is confusing to me. But it does make some kind of sense if you think about it this way: "How many twos can you put into four?"