NYC really had two classes of work equipment. One group included self-propelled machines like tampers, ballast regulators, etc, that did not move in trains. The other group included anything with standard couplers and or air brakes, that could be moved in a train or could propel a train. That included locomotive cranes, snow plows, Jordan spreaders, air dump cars, etc.
The second group, car-like equipment, were simply numbered in the X-series. That included any non-revenue equipment like gondolas transferred to M of W service for welded rail trains, and crane idler cars. They were mostly maintained by the Mechanical Department.
Burro Cranes were numbered in an X- series, with a letter for the district following. They were maintained by the Work Equipment repairmen, and were overhauled at the System Work Equipment Shop in Jackson, MI along with all the equipment in the first group.
The X-398E Burro Crane was on the Eastern District in Watertown, NY. W was Western District, B was Southern District (but they had some W machines there, left over from when the Ohio Central belonged to the Western District), and M was the Northern District. The letters actually indicated Eastern, Western, Big Four and Michigan Central. By the time I was in the M of W Dept in 1963 the District designations were not important, and machines were moved over the system as needed. New York District never got a distinguishing letter, because by the time it was split off from the Eastern District it didn't make any difference.
I don't have a reference for it, but I believe the X-equipment was generally classed in lot numbers, the same as revenue cars.
As another data point, in 1966 when Dewitt was clobbered with 52 inches of snow, we had three Jordan Spreaders there to plow it out. They came from Kingston, Watertown and Ashtabula, and were numbered X-6312, X-6313 and X-6314 with no District letter following. The consecutive numbers were coincidental.