There is Middle America and there is Central America.
Middle America is a relatively vague term referring to the places in the middle of the Americas including Mexico, the Caribbean, Central America and sometimes, Colombia, Venezuela and the Guianas.
Central America is a much more limited term which specifies the countries of Guatemala, Belize, El Salvador, Honduras, Nicaragua, Costa Rica, and Panama. In other words, the land mass between Colombia and Mexico.
So Central America does, in fact, not include Mexico.
First of all, I've more often seen "Middle America" refer to the American (United States) "heartland"- from about Colorado in the West to Ohio in the East. Though I looked it up and also see that the term is used the way you did here.
Second, Central America is just as vaguely defined as your definition of Middle America. Google "Central America map" and you will find about a 60/40 or 70/30 split between those that include Mexico and those that do not, with a majority reflecting your definition.
Edit: ow (that is what everybody is waiting for ofcourse... why else read the comments.... :P ) I got 9/24