Very different indeed, to the point that typical Muslim expositions on the matter are rather confusing.
Idolatry involves actually perceiving something as divine or particularly connected to the divine when it is not entitled to such a perception. Of course, that leaves hanging the matter of who has the authority to decree something as "properly" divine. For most people beyond Muslims and (some) Christians that is basically a free choice.
Polytheism is something else entirely: the belief in several different deities that are not always avatars of each other.
Idolatry implies a mistaken or conscious choice to raise something non-divine to divine status. Polytheism is quite unrelated to it.
As is Paganism, which is one of the first few beliefs and actually a much healthier faith than one would assume from reading the Qur'an.
But in that case, isn't idolatry always relative to religious perspective?
Who says, "I am engaged in idolatry"?
Isn't that like saying, "I am lying".
Like saying, "I have raised this thing to godly status despite it not being godly in status. I am worshipping it as if it were a god, even though I know it is not a god."
In Islam, there is The God, and then there is everything else. That doesn't mean people can't be close to The God (like the Prophet is close, for example), but they can never be The God.
In polytheism, there are many gods, and all of them are worthy of worship in their eyes. So they wouldn't say, "I am engaged in idolatry." It wouldn't make sense to them because they haven't made the distinction.
I think the question of "idolatry" is a call to question what you have truly devoted yourself to and ask if it is an invention or a created thing.
Atheism, of course, is unrelated and even opposed to both polytheism and idolatry, yet apparently it also falls under "Shirk".
It seems that most Muslims are literally unaware of those concepts and their own nuances, shoving it all under "Shirk" and sometimes even presuming some degree of desire to leave them behind in favor of Quranic monotheism.
The concept of shirk is the avoidance of responsibility. In Islam, Muslims have a responsibility to The God. Disbelieving in The God is one way to fail to fulfill that responsibility. This seems apparent. What nuance is being overlooked here?
If nothing else, that indicates a huge degree of difficulty in understanding religion.
I suppose this often happens whenever someone from one religion projects their meanings and values onto another religion and treat them as if they were all the same...
They do not, unless we want people to actually attain mutual understanding and therefore have a fighting chance of mutual respect.
Which I think is a worthy and necessary goal, but I guess many will disagree.
I don't see the elimination of contextual definition as a worthy and necessary goal. Why doesn't it suffice to note the differences?
I think you are grossly understimating the true reach of that divergence of meanings. It makes understanding and respect of Muslims by non-Muslims much more difficult than it could otherwise be, and all but forbids Muslims from understanding religion in general.
Okay.
But if you merge meanings and eliminate contextual relevance, people will just invent new words to describe what they meant before meanings were merged and all of the obstacles you think you have removed will remain.
If Muslims stopped using the word "idolatry" to refer to what other religions are doing, they would use or invent a new word (I'll call it "iconlatry") to describe exactly what they meant when they said "idolatry" before. And vice versa for polytheists, pagans, non-muslims, et al. At which point, it's possible that the word "idolatry" loses all meaning and becomes an obstacle to understanding in and of itself.