... If e.g. someone looks at the Turner painting and is "not moved"... then the axiom holds that... it is not in fact art...
Correct... for that
viewer, that is.
So, if someone thinks it is and someone thinks it isn't (art), on an individual and obviously subjective level, how do we come up with an objective definition (which seems to be what are you driving at)?
Well... we can not.
It remains deeply subjective. What humanity "objectively" considers art is a historic and group consensus of those who subjectively think it is. In other words, if there are enough people, across social groups and time, that consider Turner an artist, then he is.