Lots of good answers here. I'll just add some detail.
First, some nomenclature clarification. The Model 14 continued the line of the K-38 Masterpiece, which started in 1947. "Target" was never part of its name. The 4" version, introduced in 1949, was the .38 Combat Masterpiece (not K-38 Combat Masterpiece), later it was given the Model 15 designation.
As mentioned, the tapered barrel on the K-38 Masterpiece was dropped in favor of the K-38 Heavy Masterpiece in about 1952. The Combat Masterpiece continued to have the tapered barrel until the mid-1980s.
Let's set the cutoff for this discussion at 1982. After that, a lot of things changed in the K Target line. The K-38 Masterpiece was discontinued for nine years, and the .38 Combat Masterpiece more or less replaced it when, in 1986, the 6" and 8 3/8" barrels were added to the Model 15 line.
Until 1964, the .38 Combat Masterpiece had a 4" barrel, notwithstanding some special orders. In 1964, during the Model 15-2 run, a 2" version was added.
Until 1959, the K-38 Masterpiece had a 6" barrel. In 1959, the 8 3/8" barrel was added. Again, there were special order exceptions.
Especially in the early years (before model numbers) there were some exceptions to the sight arrangements. But generally speaking, the K-38 Masterpiece came with a Patridge front sight and the .38 Combat Masterpiece came with a Baughman ramp front sight.
The original postwar plan was to separate the K frame target sight guns from the fixed sight K frame in all three chamberings, .22, .32 and .38. Prior to the war, target versions were just that, target versions of the Military & Police revolver, except for the .22, of course. So, a K-22, K-32 and K-38 were offered in the Masterpiece line with 6" barrels. Then, 4" versions were to be released, and these would be called the Combat Masterpiece. Marketwise, it didn't work out for the .32 Long, so very few of those were made. But the .22 Combat Masterpiece and the .38 Combat Masterpiece were both a huge success.