Yes, and in football the BE is most definitely a Mid Major. But there's been some pushback on this distinction of late. As I recall (dimly) the term was coined in the 90s (?) to refer to everything outside the Power5. Power6 emerged to try to stretch the canvas a little. And as I mentioned above, the distinction isn't just about quality of play, but also program funding -- coaches, facilities, athletic and academic support staff, etc.
In Men's BB, as it happens, the term has been co-opted and the Big East is not at risk of being mistaken for a mid major. In Women's BB (where this question started in this thread) the term has currency too, and the situation is not so clear. The term Power6 would be useful to cover where exactly the BE stands. This year, and maybe last, it hasn't been so easy to distinguish us from the Big10 or the Big12 or even the SEC in terms of stratification of quality play.
"We will now adopt a generic power brand in lieu of Power 6, or P6, branding," Aresco wrote in a letter Tuesday.
theathletic.com