Dark Ages - 1: the period in western Europe between the fall of the Roman Empire and the high Middle Ages, c. AD 500-1100, during which Germanic tribes swept through Europe and North Africa, often attacking and destroying towns and settlements.
a period of supposed unenlightenment: the dark ages of racism. (the dark ages) [humorous derogatory] an obscure or little-regarded period in the past, esp. as characterizing an outdated attitude or practice: the judge is living in the dark ages.
2: [Archaeology] a period in Greece and the Aegean from the end of the Bronze Age until the beginning of the archaic period. There was no building of palaces and fortresses, and the art of writing was apparently lost.
The question about capitalization does actually matter, because it serves to distinguish between a particular, named, defined period as opposed to more general descriptive terms. To me, 'High' fits that category, the central period that everyone can agree constitutes the core; but 'early' and 'late' don't, because they're not as narrowly defined, with considerably more disagreement about exactly when the medieval period starts and ends. They are descriptive terms, not a fixed part of the name.
So I think the Chicago Manual is right to point out the distinction. IIRC, the Columbia Encyclopedia is actually sort of a latecomer, and not a particularly comprehensive encyclopedia, more a quick one-volume desktop reference. If the less careful practice occurs in many other reputable sources, then the over-capitalized version could certainly be listed as a variant on the same line. But I still think that it would be better for LEO at the very least to give the preferred version first.
However, if you just wanted to treat all three the same, without regard for that distinction, it would actually be better simply to leave all three adjectives uncapitalized, as in e.g. NOAD: early Middle Ages, high Middle Ages, late Middle Ages. I could support that far more easily than unnecessarily capitalizing the descriptive adjectives.
I have similar reservations about capitalizing Dark Ages in too many other contexts. Equating it with the early Middle Ages was probably a mistake on my part, based on checking hastily in Pons-Collins yesterday, which translated it as das frühe Mittelalter. But that quote about after the fall of Rome makes sense to me (though Charlemagne seems too early on the other end), and that's really well before the Middle Ages, isn't it? So I guess the Dark Ages is actually still a useful term, even though we no longer consider it 'dark' in the sense of 'benighted,' because we don't really have anything else to call that period.
Anyway, I would certainly agree that there have been dark ages or periods at many other times in history, but in English at least, that one is the one that's normally capitalized and treated as a proper name, in the plural.
However, there does seem to be one other named period that I wasn't aware of, which is the archaeological one in the singular (?), as in one of Selima's examples:
>>The Late Bronze Age: 1300-1200 to 700-500 BC
The Dark Age: 700-500 BC to 200-150 BC
The Iron Age: 200-150 BC to 450-500 AD
If that too is generally accepted as a known, defined period, then of course it should also get a separate listing, and a marking as [archaeol.] instead of [hist.] It might be useful to see a few more examples to see whether the singular is really more common, or whether plural is also used.
The one example that looks sort of off to me is the one where it's capitalized in English even though it's used with the indefinite article.
>>This is thought to have precipitated a prolonged Dark Age marked by cultural decline and ethnic strife during the early centuries of the Iron Age.
To me that's just awkwardly written; it would have been better to rephrase to either 'the Dark Age' or 'a dark age.' (Or perhaps even 'a so-called Dark Age'; hmm ...)
One question: If you all are voting for 'dunkel' as the adjective, how do you feel about the existing entry 'finsteres Mittelalter'? Which is better, or are both used?