Real broad and rough overview (and I'm not looking this up, so bear with me if I make a minor mistake or three,
)
First practical system is the matchlock. Smoulding "match" (treated cotton "rope") thrust into the priming powder embers ignite priming pan powder, which fires the main charge in the barrel.
First major upgrade the wheel lock (which is a type of flintlock) where a spring loaded wheel spun against the flint to spark the pan. Complicated and expensive, but did work. Later a simplified mechanism (snaphaunce") did the same job and became the flintlock mechanism we recognize today.
Later on, several others types of primers were tried, including a "tape primer" very similar to a kids cap gun, eventually resulting in the percussion cap system which came into use in the early 1800s.
Other ideas came and went, the Needle gun was called that because the firing pin was a long needle which pierced the paper cartridge to strike the primer which was located on the base of the bullet.
Self contained metallic cartridges show up, and by the time of the US Civil War are in limited use, and are rimfires or pinfires.
Development continues, and we get center fire cases with internal primers, and eventually becoming the center fire external primer we use today.
Things continue, and by the 1870s you can see cases that are the origins of the "standard" sizes we use today,
As already mentioned, in an ironic twist of fate, Berdan was an American, and Boxer was a European (British, if I remember right) but Europe adopted the Berdan system and the US adopted the Boxer primer system.
Look at the origin dates of various cartridges and what size primers they used. Size of the case mattered as much as anything. Smaller calibers coming into use in the 1880s-90s and on up used smaller primers so eventually we have two sizes (diameter) small and large for both pistol and rifle rounds.
I do not know when magnum primers became a thing, but the first rifle round that had Magnum in the name was the .375 H&H in 1912 and the first pistol round named Magnum was the .357 Magnum in 1935.
Rifle primers, intended for igniting larger amounts of powder and at higher pressures usually had thicker cups and more priming compound than pistol primers. Modern large rifle primers are a bit "taller" than large pistol primers, so usually cannot be used in their place.
Small rifle primers are physically the same size (including height) as small pistol primers, but usually have thicker metal cups and more priming compound. BUT, this varies with the manufacturer.
Small rifle primers will set off that hard to ignite powder in your .357 Magnum just fine. IF they go off. Some pistols do not strike the primers hard enough to reliably ignite small rifle primers. Many will, some won't. You need to test in your gun to see.
Begin at the starting load, or under it, (if the powder maker's recommendations allow for that) and load a few test shells. Fire them an see what your results are. Proceed carefully, working up slowing using the usual steps until you get the load you want.
IF you get pressure signs on the primer, case, or (God forbid, the gun) STOP. No matter what the book says is safe, STOP there and go no further. Pressure signs are not reliable indicators of what the pressure is (in psi, or any other numbers) but they ARE a reliable indicator that your gun doesn't like something about the combination of components your are feeding it.
I know things are in short supply right now and likely to continue to be for some time, but have you considered trying to find someone who has small pistol magnum primers and will trade for your small rifle? Or trade for something else??
