>But even if that is true, it doesn't mean that there can't be a precise definition of how the feeling of consciousness arises in humans,
You could say this about "American" as well. Trying to pinpoint the formal definition of "American" is likely possible but also completely pointless because it's an arbitrary made up concept. Just like consciousness.
As a user of the word. You don't even know what it means. All you know is that when you look at something, your brain automatically tells you what is conscious and what is not. Your brain does the same thing with "American." Attempting to formalize this category is pointless.
>what properties of our minds make us think that we have this property and that it is important.
The property in your mind that does this is the same property that effects the word "American." It is the recursive nature of consciousness along with the ambiguity of the category that imprints the word with a false sense of importance.
The reality is, consciousness is simply a line of demarcation between calculator and computer or bug and human. Somewhere there is a boundary where we classify something as conscious or not conscious. What's actually going on is a gradient, and the hard boundary is made up by humans. There is little fundamental importance where you place this boundary at all.
For example are humans more conscious than a bug? Is a bug more conscious than a Machine learning AI? Is a machine learning AI more conscious than a calculator. There is very little importance in how you answer these questions. If an AI is more conscious than a calculator... so what? You arbitrarily defined your boundary to be there. You can arbitrarily define it to be anywhere you want, technically.
The search for what "consciousness" means is essentially just looking for where our brains have collectively chosen to draw this line of demarcation. It is searching for an arbitrary property that is essentially made up by the human condition.