If we're diving headfirst into realism, we could go any number of levels down the rabbit hole. First off deciding 'when' D&D is really in terms of historical reference also where and how cutting edge/obscure (I'm sure that shield lamp above really existed, but was it commonly known of, much less available and used?). Oil lanterns, though existing, were not common in much of the pre-modern world -- at least enough that the word is often associated with candle-covers instead. Of course that's in part because poor people did stuff outside during the day and wealthy people rarely had to be portable but still encumbrance-concerned and the entire dungeon-crawling-armored-knight schtick of D&D is inherently ahistoric yada-yada-yada.
But let's assuming this is a ever-so-vaguely 'medieval + renaissance (possibly - gunpowder)' world that still has Coleman-style oil lanterns. Also let's assume there are knights in armor that want to use them in D&D-style dungeons, but also keep both hands free for fighting or trap-finding or whatnot. The 'realistic' answer probably isn't hanging it from their belt or an elaborate gyroscopic gimballed setup, but instead to hire a bearer to carry it for them.
I know the reason most won't want that -- they're either an expensive and combat-worthy hireling you'd rather have fight alongside you or they are low-level/HD/CR and die in the first encounter. However, if we're doing it 'realistically,' I don't think the bearer would be more fragile than the elaborately engineered lantern options otherwise discussed. Certainly not the flame in the lantern (which I assume you need to fight, since if you have darkvision why are we doing any of this?).