As far as our sniper rifles in 7.62 NATO, the M-21 is the original sniper version of the M-14. Back in your day it came with the ART scope, then the ART II scope. Trained with them when I went to sniper school, just for familiarization.
M-24 SWS (sniper weapons system) is what I mainly used back in the day....that day was a little less distant than yours. It's a modified Remington 700 bolt action. Long action, to be able to rechamber some for .300 win magnum, bull barrel, adjustable stock, Leupold M3 Ultra 10x mildot scope. Marine nomenclature was M40, Remington but different setups.
M-25 SWS was the upgraded sniper rifle based on the M-14. It was put into service a few years after the M-24, not as a replacement but to augment or for more specific SOF missions, as it was developed for Army Special Forces & SEALs.
M-2010 ESR (enhanced sniper rifle) is the upgraded version of the M-24 chambered for .300 win mag and skeletonized stock.
M-110 SASS (semi auto sniper rifle) is based on the M-16 action, made by Knights armament. 7.62 NATO, 10 or 20 rd mag.
The last 2 are current issue ones, although I hear that H&K now has a dog in the fight and may have been contracted for a new sniper rifle.
Steyr came out with this baby a few years ago...
This is what it shoots (.308 on bottom)
It also doubles as an anti-tank rifle.
PS: It must have been the M-25 SWS I was thinking of.
I feel like I need me some of this piece of awesomeness.
Kinda reminds me of this old school shooter.