Before having done any heavy research into the subject this is my initial thoughts about the difference between Mountain, Field, and House. Namely, they are 3 different ways to emphasize the 500 by 500 cubit area of the Temple Mount.
The outer perimeter of the Temple Mount is a place for prayer, where even Gentiles are permitted to go there and prayers are more often answered favorably in that area than in other areas.
Field represents those areas of the Temple, where the Gentiles can not enter, but we can do things there on their behalf such as offering sacrifices for them.
House represents the Heichal building - where activities of holiness were done there exclusively for Jewish concerns, such as offering incense, showbread, menora, Yom Kippur service in the Holy of Holies etc.