I think a LMG is something that can be carried and utilized by infantry(historically a machine gun squad, who had different roles and carried different pieces or ammo) whether held or mounted, whereas a HMG is larger and more likely mounted or fixed and less practical for a squad of soldiers to even deploy.....
But I dunno.
English
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It is called a heavy machine because it is a heavy weapon and it's a machine gun so hmg.
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Your are absolutely correct.