Power generation facilities are typically quoted in terms of Watts, not Watt-hours. From an era where these were fossil fuel or nuclear plants or dams that provided a pretty steady level of energy. It indicates what the generation facility/asset can produce at any given moment, although if rivers run dry hydro output can decrease.

I think the numbers you cite in [0] are in terms of total annual consumption.

The (nameplate) capacity is in watts. Its production is in watt hours. Some parties have intentionally confused these two concepts (capacity and production) to make certain power sources seem better than they are and other power sources worse than they are. The media consistently and clearly intentionally confuses these two concepts to prevent most people from learning how bad the numbers really are.

PS Capacity factor is the ratio between capacity and production. Its probably the single most important factor when comparing different generation types. Its intentionally made confusing because NPPs have 90+ cap factor and renewables have about 10ish cap factor. This makes renewables seem competitive when they are not. That's why this is always presented in a confusing manor. PPS Its also the main reason power is so expensive in CA.

It's not really anything to do with "an era where these [...] provided a pretty steady level of energy", it's just the only way that makes sense to describe it in any case.

Demand is always instantaneous, and transmission lines from a generator need to be sized for the maximum instantaneous generation, so in terms of the size of (any kind of) generator it's the main thing you're interested in. Capacity factor brings in seasonal / daily output variation but that's a whole different thing.