"Between you and me" was how I was always taught, although different countries' rules--Australia, England, and/or Canada--often differ from American English, so you may be right.

What I was taught was that, if you could replace "I/me" with an object like him/her/us/them, you should use "me." That is, you could say, "Between the two of them" or "Between us." If you could replace "I/me" with a pronoun like he/she/we/they, you should use "I." Which means you end up with:

"Jane and I went to the circus." "We went to the circus."

"The girl went to the circus with Jane and me." "The girl went to the circus with us."