Personally I have never understood why people keep acting and talking as if consciousness were binary.

I agree and I don't think that's what anyone claims it here. What is interesting is that one small area of her brain can be reversibly affected in a way that completely removes both consciousness and memory formation while keeping the person awake otherwise, which is very much different from sedation or general anaesthesia.

Yet people keep asking things like "do animals have consciousness?" and I keep wondering if I'm missing something or if the answer is just really obviously "yes to varying degrees depending on the animal but almost always less than humans do."

I don't think serious neuroscientists ask questions like that, and yes, the obvious answer better be correct.

From a medical perspective, the question of whether rather than to what degree a patient is conscious has serious ethical/practical importance. Basically, to treat a conscious patient you need his/her consent, for an unconscious one you don't, because you're excused for not getting it. So that variable, in a binary sense, is something every medical researcher is very aware of.

I agree that it is not ideal as a measure of how well someone is able to integrate information into his/her subjective worldmap (or however you want to define consciousness), but it is very available.

There are at least two distinct senses in which consciousness can be binary. The first sense is the kind you are probably thinking about: the range between e.g. insects, dogs, and humans, or maybe between early and late-stage Alzheimers.

The second sense is the kind that your interlocutors are (I surmise) thinking about. Imagine this: A being that is functionally exactly like you, and that is experiencing exactly what you are experiencing, except that it is experiencing everything "only half as much." It still behaves the same way as you, and it still thinks the same way as you; it's just that it's thoughts only count half.

If this sounds ridiculous to you, well, then you agree with your interlocutors. :) Personally, I think that there IS such a thing as partial consciousness in the sense described above, and I can link you to literature if you like.

EDIT: The place to start is Nick Bostrom's "Quantity of Experience: Brain Duplication and Degrees of Consciousness," available for free online.

EDIT: But the people who ask "Do animals have consciousness" are probably talking about the first kind, in which case I share your frustration. The second kind is more what people talk about when they ask e.g. "Could a machine have consciousness?"

Right, it seems similar. However, the memory of absence seizures is partially retained, so consciousness is not necessarily completely off:

while verbal responses were very often interrupted during seizures, some patients recalled questions or commands given during a seizure, and responded appropriately after the seizure ended

See also this review:

electrical stimulation of the claustrum, in a conscious patient with intractable complex partial epilepsy undergoing a preoperative investigation, produces a clinical state of immobility and complete lack of interaction with the environment, but retainment of some automatic movements, quite similar to cases of partial complex seizures.

There are speculations, certainly. But so far this is one experiment on one epileptic patient with a piece of hippocampus removed in an attempt to control her seizures. This is a long way away from reliably and reversibly switching off consciousness and memory formation on demand.

Is that actually the same thing? I was under hypnosis once, and it wasn't that I wasn't thinking at all, just less. I would expect not conscious to mean I had no awareness of what is going on.

I don't think this is the same. There is no memory of the event with the electrical stimulation and my (very limited) impression of hypnosis is that there's nothing about the process in itself that stops you from being able to remember.