I feel it is inevitable in any class that remotely pertains to some form of art that there should be at least one conversation about how we define said art form. I have had this conversation multiple times about poetry, as well as sculpture, music, and capital-A Art in general. Said discussions usually begin with a simple question, “how do we define X?” and pass through increasingly broad definitions until they reach the inevitable conclusion: “the world cannot be categorized and to attempt to do so is a pointless and/or counterproductive endeavor.” I love this conversation the same way I love walking around my house: it’s safe and familiar, and nothing can surprise me in there.
I think, and maybe this is cynical, but I think the impulse to reject categorization often comes from the difficulty of categorization. You spend enough time trying nail down exactly what is and isn’t art, you’re eventually gonna figure out it’s really hard to do. I think it’s reasonable, once you’ve figured out how difficult a question is to answer, to consider the possibility that the premise of the question is flawed. But I think this question, “what is art?” is actually pretty useful, which is probably why it keeps coming up.
I think we can all agree that categories are mostly a social construct. A tomato is a fruit and a vegetable. A stool may or may not be a chair. But social constructs exist for a reason, and we can acknowledge their limitations without throwing them out completely. Categories allow us to group things with common traits, talk about things as a whole instead of individually. If we had to give a new name to every object that existed communication would be impossible. “I ate glee-glum when the Sun was high and the next time the sun is high I will eat gooah” (glee-glum and gooah being the names of two distinct apples).
The definitions of art are tenuous, blurry, and subject to change and individual perspective, and it is not just in spite of this but because of this that we have to have the “what is art” conversation over and over again. To effectively communicate about art, a group of people have to come to some kind of collective understanding about what art means to them.
Post-Modern American Fiction 
I really like this post, and I think it presents an interesting discussion on the constitution of art. I also agree that art should be interpreted and understood through what it means to the viewer, but I’m not sure there necessarily needs to be a consensus about this; different pieces of art will mean different things to different people, and I don’t think it these particular interpretations are any more or less valid. I feel art is highly individualized, which returns to another point of yours: categorization is difficult, and it’s easy for us to want to reject it because of that. Art is tough, man.
I agree that it is incredibly difficult to provide concrete parameters to the category of art, and I can’t help but think of the creation of a law as a good comparison to this concept. Obviously we cannot write a perfectly air-tight law as there are constant innovations and unknowns in the way that life develops that cannot be accounted for, and it is because we cannot write a black-and-white law that we have officials to provide their best and unbiased interpretation of how the law would or should not apply to a given situation (picture the positive ideal of this, not the murky reality). It is the same thing for art: art is a nebulous concept and to come up with an all-encompassing definition is impossible due to the immense possibility of things people can do. Because of this, it is up to the individual to decide what they think defines the sharp lines around the category of art, and so long as each person can express these lines and can respect that everyone else is going to have a slightly different interpretation, then the purpose of categorization in terms of the definition of art has served its purpose.