Point quibbled with...

Actually, when I saw the thread I immediately thought of Orcrist, Glamdring, & Sting. I'd forgotten about Anduril, but I'm not convinced by the quoted passage either. Here's why...
Glamdring & Sting both glow for a specific reason - the presence of orc-kind. In fact, in Riddles In The Dark, there is the suggestion that the intensity of the glow is proportionate to the presence, and also that elf-blades, in general, glow. However there is an equally strong suggestion that that is only true if they had been made
"for the goblin wars of which so many songs had sung." So yes, I can agree with First Age weapons, but it must be broadened to elven weapons. The passage includes the term "an elven blade", which cannot be used to determine whether Tolkien meant only swords e.g. leaf spears are bladed. I suggest it is likely to be poetic rather than literal, so no firm judgement can be made.
However, I don't buy into your Orcrist quote, Dal. We know that it's a sister blade of Glamdring. Therefore it glows in the presence of orc-kind, no problem.
"It is said in songs that... Nah, sorry, all foes? Blatantly poetic licence; he's admitted it straight away. So I'll stick with glowing in the presence of orc-kind.
Then there's Anduril. I would be leaning heavily on the "poetic licence" argument again, except for one thing. The Wiki entry on Narsil has this to say,
Some passages in Tolkien's writings... imply or indicate that Narsil/Andúril glowed, similarly to Sting and Glamdring. However, Andúril was Dwarven-made rather than Elven-made, so unlike Sting and Glamdring, Narsil/Andúril did not glow blue in the presence of Orcs. Rather, it glowed with a red light in sunlight, and a white light in moonlight. However, most of the quotes given are of the "like" type - it glowed "like" a flame, etc. That's poetic licence and comes down to how literally you want to take the Prof's word.
However, there's also this, in that Wiki article -
"Tolkien confirms that this glow was not simply due to reflection or polishing in a private letter, where he describes Andúril as glowing with an "elvish light". So, yes, Anduril does glow, but I suggest we cannot be sure what effect the breaking has had. It did not glow like an elven blade does before it was broken. The quotes concerning the use of Anduril (i.e. in Aragorn's hand), I do not find convincing as evidence; too much possibility of storyteller's exagerration. So is Tolkien referring to the time before it was broken, does it not glow after the reforging & he's just being poetic, or has the reforging changed the way it glows?
The whole argument here has a problem anyway...
Fact: We know that weapons of the First Age, of dwarvish or elvish manufacture, can be made to glow.
Fact: We know that much knowledge survived the destruction of Beleriand. Celebrimbor & others; Khazad-dum was scarcely touched in the First Age and, it is reasonable to suppose, as the first & greatest dwarf home, the lore there would be at least the equal of any other; never mind the fact they took in the survivors of Belegost, the neighbour of Telchar's Nogrod.
Fact: The Smiths of Hollin were active well into the Second Age, & Khazad-dum well into the Third.
Supposition: I can't think of a single magical item in any fantasy or mythological setting that does not "die" when it is broken. Ergo, if you take a "biblical" view of LOTR & take it that Anduril is glowing, it must be by virtue of its elvish re-forging, and therefore the elves of Rivendell, at least, in the late Third Age have also retained this knowledge.
Conclusion: Much as I hate to agree with Vastin, because my gut feeling is against it, making a glowing weapon is a not much greater feat than making a magical weapon in the first place, and that knowledge is available to dwarves & elves both, well into the Third Age. Although his claim that Tolkien practically invented glowing swords, I think, is complete cobblers. I'm sure that idea has been around for an awfully long time. I can't think of any direct evidence, but Tolkien created very little; he just rearranged an awful lot...
Incidentally, there's one other weapon listed as glowing ("glittering like ice with a pale light") , and that's Ringil, a sword of Fingolfin.