I think the "Stalking Rifle" reference came from the UK.

In England, shooting usually meant shooting a shotgun eg pheasants or grouse. Hunting meant riding a horse. Stalking meant walking and hunting deer with a rifle.

Hopefully nowadays they are speaking English better and hunting means hunting.

Stalking Rifle I think refers to activities such as Scottish deer stalking.

Mauser 98 in .275 Rigby. .303 Lee Speed. Single shots of various types. .30-06. 8x57. 6.5x54 M-S.

I think replicating the concept today is as described. Longish slim barrel. Medium calibre. Light to medium weight. Lower powered scope.