As a translator owner and someone that works occassionally in the translator field, I think that there are several murkey areas of translator operation that tend to confuse people because the rules vary by situation.
First, if a translator is a fill-in. the AM or FM primary station does not have to own the translator, but they can. They can also lease time or work out any other deal that's mutually acceptable. The major caveat is that whatever translator they use, the translator's 60dBu contour cannot exceed the daytime service contour of the primary station (1mV for FM or 2mV for AM). There is also a maximum permitted distance for AM fill-ins.
FM stations can be rebroadcast on either fill-in or non-fill-in translators whereas AM stations can only be broadcast on Fill-in translators. The rules permit fill-in translators to be:
A. Operated with larger coverage areas (greater antenna height and up to 250W) than non-fill-ins. I have seen applications for translators with 250W at 300 meters giving them coverages that rival class A FM stations. There is no height limitation for fill-in translators, but there is a coverage limitation if the translator is used as a fill-in for an AM station. That limitation is that the translator's service contour cannot exceed 40km from the AM tower, even if it is still within the 2mV contour of the AM station. That effectively limits the height of AM fill-ins to between 300 and 600 meters in most cases.
B. Reimbursed/leased by AM or FM stations.
C. Can be, but not required to be, owned by the host station
D. Be fed by other methods besides just off-air signals. In other words, you can use an STL or two Dixie cups and a string to get programming to a fill-in translator.
The current FCC rules specify that AM stations can only be rebroadcast on translators authorized prior to May of 2009. Translators authorized prior to May of 2009 can be bought, sold, leased, bartered, etc. by AM stations and these translators can rebroadcast AM or FM signals. Translators authorized after May of 2009 can currently only rebroadcast FM (including HD) signals, but that will probably change after the next LPFM window opens.
Finally, it is worth noting that translator HAAT calculations are very different than for standard FM stations. For translators, the HAAT is calculated on a per-radial basis. Operating power is specified along each radial. not as the average of all radials. In other words, if you are a non-fill-in translator, using an omnidirectional antenna, the HAAT of the LOWEST radial determines the operating power for the entire translator.
Regardless of other limitations, or not, translator coverages are limted by the need to cause no interference. In most cases, translators cannot operate with anything approaching maximized facilities due to interference constraints, so omnidirectional translators with 250W at 300+ meters are pretty rare.