In the 1939 catalog, the company calls the gun the Smith & Wesson Terrier. If I recall correctly, box ends for these guns were labeled ".38/32 Terrier". If you just register it as a Terrier, that should be good enough. S&W made only one gun with that name, and this is it. There will be no confusion. (Other gun companies marketed guns with "terrier" in the name, but they are not S&Ws -- so no conflict."
Funny about the M&P comparison. When I first saw the pictures that was what I thought too: "Round Butt M&P snubby in .38 Special, prewar." Then I read what you wrote (well, most of it, since I missed the serial number) and realized that you had a Terrier. The terrier is sort of a 5/6 scale version of the M&P (but with five shots instead of six, as was pointed out, because of the reduced cylinder geography), but it's hard to see that proportion without having the two side by side.
Additional info: the Terrier is actually a short-barrel, round-butt variant of the model that S&W called the .38 Regulation Police. That model was introduced in 1917 with its own serial number range. In 1936 S&W introduced the Terrier version with serial numbers starting around 39000 in the RP series. In 1940 they suspended production of the RP line around serial number 54,400. Of the roughly 15000 .38 RPs produced in that period, I don't know for sure how many were Terriers. But I would guess that only three thousand or so were produced. That makes your gun uncommon, but not rare. If anyone has better numbers, I'd like to hear them.
Let me take mild disagreement with the implication that the .38 S&W is not a serious self-defense cartridge. Yes, it is less powerful than a .38 Special, but the .38 S&W is no slouch in its own right. Commonwealth forces purchased hundreds of thousands of the M&P model in that caliber in WWII, and they weren't stupid about sidearm requirements. S&W notes in the 1939 catalog that the .38 S&W penetrates 4-1/2 7/8" pine boards at 20 feet, which is good enough to do some soft tissue damage even if it has to go through an overcoat, jacket and vest in order to do so. Over the last hundred years or so, all the people who had serious health issues in consequence of being hit by a .38 S&W bullet would disagree that it is an inferior load. The self-defense shooting rules are: place your shot correctly and make it deep enough. The .38 S&W can handle the deep-enough part at standard snubby gunfight distances. The placement is up to you. You would be worse off with a .44 if you couldn't hit the threat.
