Generally speaking, first hand citations are considered preferable.
Sure. But one of the images shown is, quite literally, the
actual lantern used by Guy Fawkes. In November 1605. With a link directly to the British Museum that holds it. I think that ought to be adequate evidence that these lanterns were around. Lanterns of this exact type were, in fact, worn on people's belts.
Correct. Just like plate armor.
Notably, machine tools were invented in the 17th century, which made that level of precision metal working possible (similar to the tools you need to make full plate). Not that I have a problem with early modern technology in my D&D, but there are still plenty of people like their D&D medieval.
Well, other than the (as mentioned) full plate, which doesn't come into existence until ca. 1420. So, yes, we have both physical and documentary evidence from at the very least the first decade of the 1600s, and given this was Some Random Dude having a lantern of this type, it's clear the tech must have been commonplace long before that.
I'll also note that no one bats an eye at the Bullseye Lantern being present in 5e's items, despite the fact that
it wasn't developed until the 18th century,
well after the medieval period. Hence why I (almost) never take very seriously any claim that some technology is "too modern" or "not modern enough" to fit into D&D. It freely takes things from 18th or even 19th century technology when desired, while leaving other things mired in 9th-century technology, without much rhyme or reason beyond "we wanted it". The parenthetical almost there is for like...9mm handguns or computers or whatever. Those are pretty obviously beyond the pale. But when D&D rejects cannons and hand cannons for being too "modern" despite having been developed objectively during the late Medieval period, while accepting plate armor and bullseye lanterns, which weren't developed until the Renaissance or
18th century, it's hard to take most of these "that's too modern" claims very seriously.
There is surface crude oil in some parts of the world, and I have kind of felt that was common in D&D settings. That might be showing my age though. Certainly animal fats would be more common in most parts of the real world, and they would be less hot, and shed less light, and produce a lot of smoke. The classic D&D lantern though, is anachronistically modern, most resembling a kerosene lantern, which, as several people here have experience of, burns very hot, and you would want to keep it well away from your private parts.
Certainly; if we were in fact using modern ones that would be a distinct problem.
But oil lamps have burned all sorts of oils for millennia. In the ancient Greek and Roman Mediterranean, olive oil was the dominant lamp oil, for example, and while all such lanterns will produce
some smoke, I imagine vegetable oils would be reasonably useful and wouldn't be ludicrously hot. (I'm reminded, for example, of how Apuleius' rendition of
Cupid and Psyche specifically features a droplet of hot oil leaping from her oil lamp to reach Cupid because even the oil is rashly attracted to his beauty; the oil does burn him, but Psyche can carry the lamp without hurting herself.)
There might also be sources of low-temp, low-smoke oil that simply don't exist in our world, vegetable or wood sources, in addition to possible petroleum options.