I can throw in some terminology— the one-handed post you're referring to is sometimes now known as the "bullseye" stance. Here's a picture from a 1904 US army manual, and it's still used by some target shooters today. Up through WWII, a popular technique was "point shooting," which was still one-handed, but facing the target. The idea was that people are already naturally good at being able to accurately point at things with their fingers, without even needing to look down sights or anything, and they should just do that with a gun in their hand.

The Weaver stance, where one bent arm supports the other, was invented/popularized in the 1950s, and the Isosceles stance, with both arms straight out, became popular around 1980. Both of those are pretty recognizable from cop shows, and people argue over which is better today. People also still advocate knowing how to shoot one-handed, but in a modern stance with the other hand in a tight fist, tucked in to the chest.

Part of the rationale for the original sideways stance was the same as a duel in fencing— the side of your body is a smaller target than your chest, and you want to present the smallest target. Exposing your side is actually a terrible idea for modern soldiers/law enforcement though, because bulletproof vests protect your chest best and your armpits worst. This is clear in this 1920s photo of a bulletproof vest test— the shooter, standing sideways, is less protected than the test subject, facing forwards.

That is the extent of my knowledge, so I realize that I'm breaking the "Can I answer follow-up questions?" rule of posting answers. I'm happy to delete this comment if need be.

As a small arms historian I can provide a few likely reasons that culminated in one handed pistols remaining popular. None of these, to my knowledge, have been extensively researched but by observing early handguns and the people using them it seems clear enough.

1st: Swords

Much like a fencer the idea of pistols, as gentlemen's weapon or even in combat, was to present as narrow a figure as possible to your opponent.

In fencing this also provided reach. In shooting before modern sights this same outreach would likely add to the "sight radius" of the shooter and improve accuracy. After the intoduction of proper sights on handguns the method likely just clung on as tradition.

2nd: Apathy

The truth is many pistols in military and police settings were primarily symbols of rank until about the First World War. Realistically when force was required the rifle was favored or, especially for police, a club.

In fairness most nations realized the true importance of handguns at about the turn of the century. Just before WWI we start to see .455 Webley, .45 ACP, and 9mm Parabellum appearing with an emphasis on effectiveness. The Great War took these ideas to the trenches and proved the need, in many minds, for reliable and accurate pistol fire.

3rd: Cavalry

One of the first military units expected to actually use their guns were the cavalry. From horseback, one handed shooting was best.

In fact we see early pistols made to satisfy handling and loading one handed specifically for cavalry. The French 1892 opens to the right so it could be held with the reigns while a right handed shooter loaded. The Spanish Jo.Lo.Ar actually has a handle mounted to the slide for single-handed cocking.

So when your primary user of handguns is, by nature, one handed the design follows. Other users are then forced to follow the design.

So, with the pistol becoming a tool over a symbol, the lack of cavalry requiring one-handed operation, and a growing sense of "there must be a better way" we finally left tradition for effectiveness.