To throw some other things into the mix, "sniper" is a term widely used and misused today.
About the only common thing is that (today) a sniper is someone who shoots people, from concealment, and usually at some distance.
The military has developed a very special set of requirements over the years, and often disparages anything less as not being a "sniper".
A "sniper" need not be someone who has the deep training needed to operate behind enemy lines, and wait for days if necessary to hit a specific target. Someone who can is certrainly a sniper, but many, many historical snipers do not fall in that category.
Police have "snipers", although at first they called them "counter snipers" and now generally call them sharpshooters or marksmen. They don't typically operate at the distances of a military sniper, nor is there any "behind enemy lines" type stuff involved. One can call them whatever seems best for political advantage, but what they are, are snipers.
The criminal shooter named the "DC Sniper" (who it turned out were a pair of people who justified murder with religious fanaticism) used an AR, fired from inside a modified car (for concealment), at not very great distances.
The Minutemen of the US revolutionary war were "snipers" to the British (although the British didn't use the term at the time), as they fired from concealment, rather than standing in the open, like "proper" soldiers did.
Carlos Hathcock did make a very long shot using a scoped M2 .50 caliber machinegun, but he did not use is as an automatic weapon, he fired it as a semi-auto, one shot at a time.
Haya (I understand) shot a number of Russians with a submachine gun, in addition to kills with the rifle.
Many of history's "snipers" were NOT specially trained individuals designated by their military as snipers, but were rather simply skilled individuals, often regular infantry, who became designated snipers as their skills were recognized.
Their "training" was OJT, and later served as the basis for the specialized training that has been developed and is in use now.
Another story is the one about Unertl, who was a sniper during WW I. He was so disgusted with the optics available that he began making his own. When he came to the US as a immigrant, he was asked if he ever killed anyone. Supposedly he replied, "yes, 96 Bulgarians". They let him in.