The Utah listings follow a field office by field officer format. What you are labeling a "district" is, in fact, a field office. The BLM is organized into 12 state offices, 11 western state offices and the Eastern States Field Office, which consists of the states bordering the Mississippi River and east to the Atlantic. The next hierarchy down from the state office is the district. Each district is further sub-divided into field offices.
Some field offices have field stations under them when the size of the field office (in acres) and large area (in distances) is sufficient that a satellite office is needed. In the Cedar City Field Office, there is a field station at Hanksville and the Grand Staircase-Escalante National Monument, headquartered in Kanab, also has a field station at Escalante and two visitor centers located away from the monument headquarters, Big Water and Cannonville.
In Utah, there are four districts. The West Desert District (District #1) consists of the Fillmore Field Office and the Salt Lake Field Office. The Green River District (District #2) consists of the Vernal Field Office and the Price Field Office. The Canyon Country District (District #3) consists of the Moab Field Office and the Monticello Field Office. Lastly there is the Color Country District (District #4) which consists of the Cedar City Field Office, the Richfield Field Office, the Kanab Field Office and the St. George Field Office. The Grand Staircase-Escalante National Monument has a large enough land area and workload that it is not part of any of the districts and the monument manager, like the district managers, reports directly to the Utah State Director, the highest ranking BLM official in the state.
This is important for the scanner listener for a couple reasons. First, radio frequencies and channel plans are generally organized by district, not the field office. Frequencies are allocated to each district and there might be a district wide channel plan. Some field offices are assigned their own frequency, rather than use a district wide net. In some cases there might be a channel plan or more for each field office as each has a unique set of other government agencies to interact with at the federal, state and local level. Lastly, the first number of BLM radio designators is the district number, the second is the field office or major function within the district, the third is the function and the fourth is the individual unit.
The best example I can use of this radio designator system is here in California. The state is organized into three districts, Central California (District #1), Northern California (District #2) and the California Desert District (District #6). The Bishop Field Office, one of five field offices in the district is #7. So all Bishop F.O. units are 17xx. Recreation is function #4 in California. So a Bakersfield District, Bishop Field Office recreation unit would be 174x. Positions #0 is the supervisory position of an organization. So the supervisor would be 1740 and those below that person are 1741, 1742, etc. At the district level a second digit of 2 is usually assigned to operations personnel so the district supervisor of recreation would likely be 1240. Careful listening to traffic on any one of the districts in Utah will eventually lead to understanding the first three numbers of this system.
The BLM organization is similar to the U.S. Forest Service (USFS) organization with a BLM state office being similar to a USFS regional office. BLM districts are similar to a USFS National Forests. The BLM field offices are similar to USFS ranger districts. Calling field offices, districts, is similar to calling every ranger district a National Forest.
I would reorganize what you've done based on BLM districts, with field offices under each district. The organization of any BLM state office can be found on the state office site by clicking on to the "Directory" link. The directory will show the hierarchy of the state office organization.