my understanding is the fireman's tool while similar was larger, did not have a sharpened edge on the blade as it was used for pulling down wooden structures rather than cutting, and so was shaped a bit differently. the jingama more closely resembles a small kama blade with a sharp chisel edge. unlike the firefighter version, the jongama had to be sharp as it was also used by the mounted warriors to cut hay for their horses.
alsom of course used to cut flesh.
the jingama was derived from the tobiguchi fire tool, so is very similar. the tobiguchi were provided to samurai and daimyos who were responsible not only for defence but for firefighting, and when attacked they were sometimes picked up by the warriors for both firefighting and defence at the same time, and later weaponised versions, jingama, or tobiguchi 'sword' were made. i would also assume if edo period, it is very late in it (ie. closer to 1868)
tobiguchi (kite's beak) fire tool and hikeshi (firemen) for comparison: