And yet, we are confronted by the fact that we use the concept of consciousness in dozens of different contexts and we understand perfectly well (usually) what people are talking about. That's the basic insight of ordinary language philosophy. These aren't fundamental mysteries, they're webs of concepts tied up in language and careful attention to how they are used, un packing the web and knot of uses, reveals what they mean. The definition is a "rule" or explanation for their use in language, which is not "subjective" (private) but public behaviour.
That's actually one of the core parts of Wittgenstein's "Private Language Argument"[0]. The problem isn't so much that verification is "difficult", it's that certainty in the specific sense of intelligibly private is a logical impossibility. The implication is that "experiential qualia" have a specific intelligible meaning in language (or even more broadly in behaviour), which is to say that our "subjectivity" is personal but not private. Further implications tumble outward. For instance, Wittgenstein spends a lot of time discussing the "qualia" of pain. It was (and still is, obviously considering the rise of "AI" discussions lately) an enormously consequential argument in philosophy.
[0]: https://en.wikipedia.org/wiki/Private_language_argument#:~:t....
"Wittgenstein does not present his arguments in a succinct and linear fashion; instead, he describes particular uses of language, and prompts the reader to contemplate the implications of those uses. As a result, there is considerable dispute about both the nature of the argument and its implications. Indeed, it has become common to talk of private language arguments."
Yeah, sorry, I don't have a hundred hours to spend on some dude who decided to write half of an argument in ten different places. Maybe a link to plato.stanford.edu would've been better.
But the wiki entry is a far more succinct summation of the main points of the argument. The very first subsection, which you apparently didn't even try to read, in the wiki is "Significance" which summarizes the argument in all of...2 paragraphs.
I think you're saying we can get a general sense of what is meant without a formal definition. That's a fuzzy definition; i.e., one with high uncertainty/low confidence. My point is that I don't know that we have the tools to make high-confidence claims. We can't even agree on when consciousness starts, whether it's a toggle switch or a dimmer switch etc. That all speaks to high levels of uncertainty, where we can't agree on a "public" language. I think the limits of language to describe the phenomena is right in line with the "we don't have the right tools" argument.