The .50 M2 Browning was not used with high regularity as a sniper rifle (with scope attached and in a semi-automatic or single-shot firing mode) out in the field very much, because as u/Daymandayman said, it was a pretty heavy beast of a gun, so lugging one around by hand to use and was set up as a dedicated, outfitted, sniper rifle while on patrol was rather uncommon. I have never heard of it being used as a "foot patrol sniper rifle" to be honest, as snipers typically carried (and today carry) a much lighter dedicated sniper rifle platform, typically chambered in a smaller round size, though of course there are .50 caliber sniper rifles that weigh a lot less than the Browning M2 that have been/are employed as well while on patrol by dedicated snipers.

However, during WWII, and both the Korean and Vietnam Wars (as well as, much more recently, Afghanistan), the M2 Browning was reportedly used as a sniper rifle at various bases and outposts, where the gun was used as a largely stationary (or at least not needing to be lugged too far) overwatch weapon set up on tripod to lay down both suppressing fire in automatic mode, as well as being used, in multiple instances, in semi-automatic mode with a telescopic sight attached as a dedicated sniper rifle.

Carlos Hathcock, the famous US Marine sniper in the Vietnam War you referenced, was the most well known sniper to utilize the M2 as a dedicated sniper platform during that war - modifying the weapon to allow for the attachment of a scope mount of his own design and firing it in anger at the enemy multiple times - including scoring what at the time was a confirmed "world distance record" kill-shot of 2250 meters. I believe (from his book and the reports like the site you linked to says), in that instance he fired at an enemy soldier on a bicycle who was transporting weapons, and the first round he fired struck the bicycle. The enemy soldier jumped off, but then went back to the damaged bike to try to retrieve the weapons, and Hathcock fired again, killing him. Hathcock (or anyone else for that matter) never to my knowledge carried the M2 into the field on patrols however for use as a dedicated sniper rifle, but just used it around forward operating bases to engage enemy troops and vehicle targets at long range, as other US Marines/soldiers reportedly did as well.

In Korea, US soldiers were known to outfit the M2 Browning .50 with scopes and use them as dedicated sniper platforms as well. This idea in Korea was said to be credited to US Captain William S. Brophy, who found the M2 .50 to be much superior (especially in high altitude, mountainous terrain) at sending rounds downrange to engage far-off enemy with long distance shots. Again though, this platform was typically not hauled around on patrols as a dedicated sniper weapon during that conflict, but rather was used around bases to engage far-off enemy who had been sighted.

To say the M2 was "commonly" used as a sniper rifle in those conflicts is probably stretching it a bit, but it was indeed employed by numerous snipers at different locations who wanted to exploit its long-range capabilities and excellent accuracy in semi-auto mode, but again was typically used as via the tripod-mounted, stationary platform located in bases or emplacements that were needing to be defended against enemy attack, and not lugged around on patrol as a dedicated sniper platform. Hathcock in Vietnam for example got the vast majority of his 93 confirmed kills using his Winchester Model 70 that was chambered in .30-06, using an 8-power Unertl scope - which many dedicated snipers used at that time (Vietnam War-era) as their primary weapon of choice.

The M2 Browning .50 has still been used very recently in fact in semi-auto mode as a dedicated sniper weapon by some US forces in Afghanistan, outfitted with a scope and tripod mount and fired from fixed locations/bunkers for purposes of long-range sniper base defense, occasionally against enemy troops and, more typically, for enemy vehicle suppression.