It's not just about the technological method of distribution, it's about the structure of the entity, how it's regulated, etc. It's a pretty standard term and while it can have other meanings and the meaning can shift (as the strict original meaning becomes more, uh, meaningless), it seems weird to complain about that usage being somehow 'all over the place'. It means something most newspaper readers understand.

Who reads newspapers these days?

> For example, the Emmy’s (and the Hollywood Reporter, for example) consider NBC, ABC, HBO, Netflix, and Amazon to all be networks on equal footing.

The trade papers may use ‘network’ interchangeably in Ye Grand Tradition of old school Deadline slang for TV, but nobody in the industry would ever confuse networks (read: ABC, CBS, NBC, FOX) with cable/premium/streaming channels.

As a mere set of eyeballs... why are they (E.g. Netflix vs CBS) different? And why should I care?

To me they are all channels. Just some don't allow binge watching or time-shifting well and others do.