Should it? Does it say that in the PH?
Yes it should, and does, and it says so in Basic Rules. Why are you even asking?
"Your Intelligence (Nature) check measures your ability to recall lore about terrain, plants and animals, the weather, and natural cycles." Camels, and other beasts, are animals. QED.
"Your Intelligence (Arcana) check measures your ability to recall lore about... the planes of existence, and the inhabitants of those planes." Celestials, fiends, and elementals are inhabitants of planes. QED. The Undead are linked to the Shadowfell, and the fey to the Feywild; whether Arcana includes lore on them is less cut-and-dried. (I'm assuming that Arcana covers *other* planes, and not all inhabitants of the Prime Material plane.)
"The DM might ask you to make a Wisdom (Survival) check to ... identify signs that owlbears live nearby..."
If you would tell me that my PC recognizes signs that owlbears live nearby, *and also doesn't have any idea whether an owlbear is Tiny or Huge*, then you can kinda claim RAW but I'm darn glad I'm not at your table.
Lore won't tell you armor class and hit dice, resistances and vulnerabilities, unless your setting includes someone who measures such things and add them to lore. It doesn't mean that the player gets to open up the Monster Manual and treat that information as PC knowledge. If that's what you have in mind, could you please say so? Because the OP wasn't about whether the PC would know stat blocks; it was about whether the PC might know the NAME of the critter they saw.
Lore DOES mean that a PC can reasonably know about dragons - to the extent that people in King Arthur's court knew what the Epic of Beowulf says about dragons, taking that as a typical source of lore for that setting. There's some important info about dragons in Beowulf: they hoard wealth and get angry when it's stolen, they breathe fire, a single dragon can kill a great warrior, etc. Anyone who's heard the Odyssey has access to lore about cyclopes. Not even a Sage always remembers everything they've ever heard in passing, but INT represents recall and education, and on a sufficient INT-based success, that lore includes: cyclopes eat sheep and humans, are descended from Poseidon and are under his protection, can throw rocks well enough to sink ships, at least one of them was easy to get falling-down drunk, that one was named Polyphemos.
"So if you use this variant, what monsters does a Guild Artisan know about? What about a Noble or Folk Hero?"
A Guild Artisan herald knows about lions, as those are monsters used in heraldry; a herald knows they are quadrupeds with fangs and claws, and associated with valor. A Noble also knows a bit about heraldry. A Guild Artisan tanner knows what wolves and bears are, if they've ever tanned those hides, or even if they've just been trained to tan those hides. A bowyer knows about monsters whose gut or sinew is used to make bowstrings. And so on. That's not from DMG, which I don't have; it would be my ruling.
"I think it is nigh-infinitely more interesting and fun to make them learn about monsters, at least those outside of the common experience of their race and culture, through experience."
That clause in the middle makes all the difference. In Forgotten Realms, goblins and giants ARE in the common experience of the race and culture of any dwarf. In 1E, *every* adventuring dwarf knew so much about giants, that they got an effective +4 AC, although the mechanic was a penalty to the giant's roll. Does that mean a dwarven PC knows that there are hill, fire, frost and storm giants? It's not in RAW, but I'd allow an INT roll, with a bonus if they spoke Giant (and thus might be aware of dialects of Giant).
Legolas recognized the balrog in Moria as a balrog. I'd bet that Gandalf had an approximate idea of its stats, and that none of the hobbits knew any lore at all about balrogs. Circe told Odysseus enough about the Sirens, that he had his crew plug their ears. Do your players prefer that none of their PCs ever know, on first contact with a monster, as much as Legolas or Odysseus did?