As others have posted, the .22/32 was the brain child of San Francisco gun dealer and west coast distributor for both Colt and S&W, Phil Bekeart around 1908.
At the time the M frame Ladysmith was a small double action .22 caliber revolver that was carried by many women in their purse and hence the nickname Ladysmith.
Bekeart approached S&W and asked if they could build a larger .22 that was more suitable for a man's hand, perhaps using the .32 caliber I frame as a platform.
Since S&W was only in the business to make money, they were not interested in doing all of the tooling necessary to produce such a revolver without a substantial order. Bekeart agreed to order 1,000 of these guns as he felt that they would sell like hotcakes.
S&W agreed to retool and to begin to produce the .22/32 revolvers on the .32 frame with a 6 inch barrel and target sights. Since in its day it was much bigger than anything S&W had built, it was referred to as the Heavy Frame Target or HFT.
As it turned out, Bekeart only received 294 of the first production run of approximately 1044 guns in 6 shipments in 1911 and most of the rest were distributed by the rest of S&W's normal distributors during 1911 and 1912.
The first 3000 guns received stocks with an assembly sequence number imprinted on the bottom of the left stock panel. This number indicates the order in which the serial numbered frame was put together and there is no correlation between assembly numbers and serial numbers. In other words they were assembled out of order.
The first batch of these fall between serial number 138226 and 139275. Bekeart did receive additional shipments after the first 6 shipments in 1911, however, there are no records of those serial numbers and only a S&W historical letter can determine if your .22/32 shipped to Bekeart. As a footnote, Bekeart shipped guns are desired by collectors and will usually command a higher price.
The .22/32 became a regular production model around 1915.
Now that you have the history, the answer to your question follows:
Once the .22/32 was readily available, many men wondered if the gun could be had with a 4" barrel instead so that it could be placed in a hunting or fishing "kit". In a way, it almost reverted the gun back to a Ladysmith for men.
The .22/32 Kit gun was born and IIRC they begin to appear around serial number 525670 in 1935. Recessed chambers appeared around 525600 so there are some later HFT's that will have recessed chambers.
When the new K frame came out around 1931, sales of the .22/32 dwindled and the gun pretty much died by the beginning of WWII in 1941.
Your model 51 was not introduced until 1960 and was built on the larger J frame that was designed to house the Chief Special and fire the .38 S&W special cartridge. The 51 also shares the serial number range of the 1953 Kit Gun which is around 52637 to 135465. The 51 is also chambered in the .22 Winchester Magnum rimfire cartridge although optional .22 long rifle cylinders were available.
Hopefully, that explains the history and development of the .22/32 and subsequent Kit Gun versions.
Oh my head, I need more coffee! 
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