> I sometimes get there by being sleep-deprived.
Ah, yes. It's a little bit different. Clarity, calm, a kind of purely-mental energy, an ease to achieve good focus. Sometimes a positive, sunny feeling (unless you're completely exhausted or stressed, in which case it's nasty); other times quite an intense but smooth joy. But at the purely physical level, batteries are running out.
The twilight state has less clarity (it's not obscure, though), and calm doesn't enter the picture. But it's full of this roiling primeval emotional drive to create (in a very wide, general sense). It's where dreams come from, and most inspiring ideas. Some folks find it easier to experience it if they linger in bed in the morning, trying to avoid yanking themselves out of sleep directly into full waking state. It's a fragile thing, handle it gently.