You're definitely right that it's important to be careful how we refer to terms like "consciousness", and your response actually illustrates part of the problem. Because people use the same term to mean two (or more) completely different things, and can therefore end up talking past each other.
For example: what I wrote was using the term "consciousness" as "being able to experience sensations", i.e. the most shallow definition of the term. Self-awareness and other "higher" mental abilities don't enter it. Almost nobody would deny that animals can experience sensations: pain and pleasure, hunger and satiation, and so on. And that shallow definition of consciousness, the one that almost nobody denies animals possess, is what I meant, and what I tried to use throughout my post. Whereas when you wrote "people possess consciousness, mere animals don't", that's a completely different meaning of the word. It's what I would call "self-awareness". And it is indeed what many (probably most) people argue differentiates animals from humans. But that's not what I was talking about. So even in a short response to a short comment, we were partly talking past each other because of attributing different meanings to the term.
It's therefore possible that I've misunderstood this guy's argument, in part. I had been assuming he meant "awareness", i.e. the ability to process sensations into qualia of experience. (Indeed, his paragraph that starts "We may balk at ..." does just that: refer to consciousness as being linked to the "feeling of a stove's heat, or the redness of its glow"). But then, earlier in his article, he says that consciousness is "a belief about what kinds of entities have beliefs, experiences, feelings, intentions and agency." Experiences are part of the shallow definition of "consciousness" (that I would call "awareness" for the sake of avoiding ambiguity), which almost nobody would dispute that animals have. But "beliefs" ranges all the way from "if I pull this lever, I will receive food" (which it can be proven that animals have) to "I believe that God exists" (which is the kind of thought that many, though not all, people would deny that animals are capable of). So his definition of "consciousness" appears to be muddy and not pinned down to a single concept.
But even if I've misunderstood what he means by consciousness, it doesn't change my argument that his logic is backwards, and that his "we should reason backwards" assertion is wrong, incorrect, just plain false. The only people for whom his asssertion "we believe they [that is, other living beings] are conscious when and because we care about them" is true are the kind of people who don't truly see other people as conscious beings but rather as mere tools for them to use and manipulate. Malignant narcissists or sociopaths, for example. But pretty much everyone who isn't some kind of sociopath reasons as follows: "this animal is conscious" (by which I mean able to experience pain) "therefore I shouldn't hurt it" (unless you have to, e.g., self-defense against an aggressive dog trying to bite you). Or "this person is a human being, self-aware like I am, therefore I shouldn't hurt their feelings" (again, the "unless you have to" exception comes up: it's not always possible to avoid hurting anyone's feelings in some complex social situations, even with the best of good intentions). Either way, whether you use the lower or higher definitions of "consciousness" (and I'm well aware that there are far more definitions than just these two, potentially muddying the waters even more), that reasoning is the correct one, and his "we should reason backwards" one is laughably wrong.
Trying to put the most generous construal I possibly can on his argument, it's possible that he meant "we treat other people as conscious beings because we care for them", i.e. when you interact with the cashier at the supermarket (assuming you're not using a self-checkout lane), you're probably not actually thinking about them as a human being, saying "Hi, how are you" and asking about their day. You're probably ignoring them, focused on paying for your groceries and going home. And they're probably doing the same to you, focused on scanning your groceries quickly to keep the line moving. From your perspective, the cashier could have been replaced by a grocery-scanning robot and your interactions (assuming the robot worked) would have been the same. If he had written that, it might have made some sense.
But what he actually wrote was that we "believe" that others are conscious when we exhibit care for them, and the natural conclusion that follows from that sentence is that when you don't greet the cashier at the supermarket and chat with them, and instead treat them no differently than you would a robot, you don't actually believe that they are a conscious being. Which is a laughably false conclusion that can be easily disproved, by simply asking people who have just finished having a cashier check out their groceries "Does that cashier have thoughts and emotions?". Pretty much everyone will say yes once they actually think about it. They believe the person has emotions. They don't always act on that belief — ask anyone who's worked in retail how many angry customers have yelled at them for something the company did, treating them like a thing rather than a human being who might not like to be yelled at — but even those angry customers, if someone were to stop them and say "Hey, that customer-service rep that you're yelling at. Are they a human being with thoughts and feelings of their own?", they would answer yes. (And some of them would then be shamed into modifying their behavior, because they have realized that their actions were not in line with their beliefs).
No. Even construing his argument in the kindest possible light, I can't find a way for it to be sensible. If he meant "we sometimes don't act like others are conscious", he should have written that. What he actually wrote is completely backwards from reality, and shows him not to be a serious thinker.