Mid-tower is a case design; don't confuse case design designation with MoBo size specs. That said, most midtowers accept ATX MoBos. Some cases on the small end of "midtowers" might be microATX only...hard to make generalizations about 'All midtowers accept ATX".
"Full ATX" is not a specification; I am fairly sure it was used to avoid confusion. There is the normal ATX, than the mini and micro ATXs.
A lot - but not all - midtowers will accept both an ATX and a microATX MoBo (they have screw positions for both).