Here's a better summary of my take... Bottom line, a DMR is modern rifleman who has shooting talents equivalent to your average shooting enthusiast, woodsman, someone how has gone through Appleseed or similar, etc, and has the configured their weapon in a manner that optimizes its accuracy... Sniper requires purpose-built weapon, a higher degree of training, and better optics... It's the difference between being able to engage bullseye first shot (properly, estimated rather than known exact ranges) and cold bore vs taking a few shots to walk rounds in, etc. (Note that Russian "snipers" are properly DMRs by proper definitions... The Russians were caught without proper sniper capabilities in Afghanistan and Chechnya having only a squad marksman, spun up SVDs and additional training but still well short of sniper as the US military defines at least until the past decade or so when Russians have modernized their military and moved away from Cold War massed tactics -- similar story with Russian Special Operations as well, where US Spec Ops forces have provided a lot of training and modernization to those units up until the past few years with Russian aggression stopping most of the military exchanges and collaboration).
Here's a more complete summary:
DMR
-More accurate than standard issue via build specs, optic, or both
-Capable of being a general use military rifle/carbine, so semi-auto required
-Able to reach further than standard issue weapon through caliber, cartridge, optic -- requires at least one to be superior than standard infantryman, but could include all
-Accuracy requirement is minute of man through expected engagement ranges, as much for accurate suppression fire at distance as for precision targeting
-Primary role for DMR soldier is in an anti-personnel role
-Works within standard infantry platoons/fireteams
-Low magnification optics are sufficient here
Sniper
-1 MOA requirement throughout intended engagement range
-A purpose-built weapon, so can be semi-auto or bolt action
-Greater accuracy requirements requires a much higher standard of marksmanship, so typically requires .308 or greater caliber in military (police snipers with shorter ranges can be well served through intermediate calibers such as 7.62x39 or 5.56 however), match grade ammo, and higher magnification optic with mil or moa dot recticle to best utilize sniper precision and skillset
-Accuracy requirement is 1 MOA noted earlier and intended to be capable of achieving objective throughout engagement ranges with just one shot
-Primary role is to disable high value targets -- officers, NCOs, other leadership, support weapon teams (MGs, grenade launchers, mortars, etc), vehicles, light armored vehicles, unarmored infrastructure, etc
-Primary role is as an independent unit, often
-Attached at company level or higher from company or battalion level HQ or Sniper companies/platoons
-High magnification optics are required and proportionate to ranges and capabilities of rifle and caliber and intended use