This gets into how cartridges are named, which also gets us into the territory of “the great thing about standards is there are so many to choose from!”
There are several different naming standards for rounds. One of the older ones is the caliber plus the year the cartridge was introduced. An example of this is the .30-06 round (.30 inch diameter, introduced in 1906), which was developed for the Springfield Model 1903 rifle, and leads to the obvious question of why the rifle is dated 1903 and the cartridge is dated 1906 when they were developed together (they tweaked the 1903 cartridge a bit). Another standard is the bullet diameter and the powder load, such as the .45-70 round (aka .45-70 government), which was .45 inch diameter bullet and a 70 grain black powder load. Yet another standard is to specify the diameter in metric along with the cartridge length, so the .22LR round is 5.6x15mmR, which means 5.6 mm diameter (.22 inches) with a cartridge length of 15 mm, and the R means “rimmed”.
A lot of European rounds are 7.62mm in diameter, which seems an odd number. Typically folks like even numbers like 9mm or 10mm. So why 7.62? Because .30 inches (a fairly round number) is 7.62mm. 5.56mm is .219 inches, effectively .22.
5.6 vs. 5.56 gets into another naming convention. That difference of a hundredth of an inch is probably below the manufacturing tolerance of the round, but they give them a different number just to differentiate them. They are both effectively .22 caliber, but they are very different rounds. Another example of this is the .357 magnum round, which is a more powerful round based on the .38 special round. The .38 round uses the neck diameter to name the round and the .357 uses the bullet diameter so that you have two different names to avoid confusion. Both rounds use a .357 diameter bullet and both have a .379 inch neck diameter. Also note that .357 inches is effectively 9mm.
The long and short of it is that naming conventions are a disorganized mess of multiple standards.
The .22 round used for shooting competitions was developed in the late 1800s. They called it the .22 “long rifle” (abbreviated .22LR) to distinguish it from an earlier .22 round that was introduced in the 1850s. The older cartridge was then renamed the .22 “short”. Folks that use the metric system call it the 5.6x15mmR (5.6mm diameter, 15mm lenth, R for rimmed). It’s low cost, low recoil, and good accuracy have made it the standard for recreational target shooting, and it is therefore no surprise that it became the standard for many types of competition shooting as well.
.22 short isn’t a very popular cartridge these days, though it is still in production. When most people say .22 round, they mean .22LR. In European competition shooting they call that same round the 5.6mm round, though upthread it was mentioned that they also use the more common .22LR name.