Generally speaking, "large format" refers to anything 4x5" up to 11x14", (and larger), which is "ultra large". "Medium format" is smaller than 4x5" but larger than 35mm. The 35mm format was called "miniature" for some time after its debut, because it is so much smaller than the sizes common at the time. The really small formats, such as those used in the "Minox"-type cameras are "sub-miniature".
I don't know where in this hierarchy 127 film fits, because although it is larger than 35mm, it is not as large as what I normally think of as medium format.
Medium format need not be 120 roll film. There are still people who shoot
2 1/4 x 3 1/4 (6x9 cm) sheet film.