I understand your definition and position. I personally disagree that art can be made accidentally. A sunset is pretty -- it's not art. But I also understand the issues with trying to discern intent.
I'm using 2 techniques here in my debate (and i am not a debate expert person).
CamelTentNosing. Once a camel's nose is in the tent, the whole thing is in the tent. So once you agree part of it is true, there's more that truth applies to.
Group Replacement. Some recently dead philospher guy taught that for any law about a social group, if you shuffle up all the social groups and your own social group was in the list and COULD be in that list, and you would not like your group to be drawn, then you had a bad and biased law.
If Danny and I go back in time when people didn't think much of what black people could accomplish, and he wrote all my papers with my name on them, and everyone thought they were great. i've misrepresented his work, and gotten a relatively unbiased response (because they were comparing my papers to my similarly colored peers).
If I revealed that Danny was writing my work all along, and aside from their anger at being duped, they then said all my work was crap, they are being disingenuous. The work was good on its own merit, regardless of the source. you can't say X is good because of the source. X is good on its own merit.
So, if my dog spills some paint and it results in a pretty cool looking painting, and I misrepresent how it was generated, but simply display it, promote it, sell it as a work of art, if you accept it as art, then it must be art.
If I dig up some gold, steal some gold, or recombine protons, neutrons and electrons from lead to make gold, it still sells for the same value in the market. It still makes the same prety jewelry.
The means of generation can be detached from the value intrinsic in the object itself.
My theory then, is if I trick you into liking it as art, you can't take it back. It's art. therefore, the source of the art isn't a factor in its determination.
If this theory is true, it would modify my logic statment to:
If it is artificially generated and beautiful, it must be art.
artificially generated still being a imprecise term. A truly natural event like a flower or sunset directly viewed wouldn't count, but a photo or painting would.
What your dog does naturally in the backyard probably wouldn't count (maybe she could dig a really cool looking hole...) but what your dog does while interacting with unnatural elements might count (rolling in paint on my drop cloth).
the beautiful clause is also odd. Obviously we allow for the beholder's eye, but in my original concept, beauty was intentionally added as part of the man-made nature of the obhect. Whereas, obviously, my dog doesn't care about that.
That can mean that beauty really means "appeal" rather than intent. The dog-made painting that I said i made appeals to you, therefore it is art because I am lying. at most, there's my Intent to present this as art rather than a big splotchy mess.
So human intent to apply value to the object and human appeal for the object are probably required components to qualifying as art.