And yet many non-Byzantine consensus protocols are equipped to handle the network fault model, which could be seen as equally Byzantine under this definition.

The problem is really that many formal proofs of consensus have focused only on the network fault model, and neglected the storage fault model.

Both network/storage fault models require practical engineering efforts to get right. I think a better term for this is “near-Byzantine” fault tolerance. It's what non-Byzantine fault tolerance looks like when implemented correctly in the real world—the GP comment is a great example of how to approach and think about this from an engineering perspective.

I dive into this in detail also here: https://www.youtube.com/watch?v=rNmZZLant9o