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There is no fixed answer, as 45 Auto and 308 Winchester, while both originally made for Large Pistol (LP) and Large Rifle (LR) primers, respectively, now have some brands of brass with pockets for small primers. This has come about for reasons to do with lead-free primers in the case of 45 Auto and because some match shooters and testing showed SR primers giving slightly more consistent velocities in the case of the Lapua 308 Winchester Palma cases with the small primers. So, you always need to double-check your particular brass.
In general, the primer pocket is usually chosen to fit the case head. Heads that are about 0.425" in diameter and smaller just in front of the rim will usually have small primer pockets, and larger-diameter ones will usually have large primer pockets, but there are exceptions, like the 454 Casull Magnum, which is a revolver cartridge that uses small rifle primers. One quick way to check is to look up load data and see what type it lists and use that.
Hodgdon is a good source of that information. Use their site to look up any load of the caliber you are interested in, and it says what primer they used.
There are three sizes of primers. SR and Small Pistol (SP) are the same physical size but are not meant to be interchangeable, though some have successfully interchanged them in some circumstances; the general advice is to use the type your load data used, as changing primers can affect the pressure in some instances. LR and Large Pistol (LP) are the same diameters, but the LR primers are about ten thousandths taller and stick out of an LP primer pocket slightly when seated. In general, SR and LR primers have thicker cups than the SP and LP primers and are harder to ignite.
Magnum primers make more gas than standard primers. A primer has to partially pressurize a case so the powder burn will sustain, and the larger the case, such as a magnum case, the more empty space has to be pressurized, so hence the term "magnum" primer for primers that make more gas. However, a standard case can also have extra space in it if the powder charge doesn't fill it very well, so magnum primers can also be helpful in that circumstance. They can also be helpful with powders that are harder to ignite due to heavier deterrent coatings. The St. Marks spherical propellants market as Winchester Ball Powders are an example, as are their Hodgdon equivalents.
This article will explain some of this.