Definition has to be "specific" as to include only objects intended. But you _intent_ remains subjective.
Definition that includes more objects than intended is not "unobjective" it is just unsuccesful.
There are more other criteria which makes a definition useful in scientific threatise, and which cause a particular diffinition to become common. The final test for effectiveness of particular definition is its wide propagation and common usage in discussions. In retrospect it is possible to analyse why some definitions become common and other did not, and come up with list of criteria for success. But the final test of usefulness is also subjective, and our list criteria will only give guide-lines but no guaranties of success. In some cases success of particular definition is just a matter of "first inventor" of certain relationsheep giving it a definition, even though later when relationsheep is better understood a more specific definition could have been given. In other cases it is just luck or higher persuasiveness of particular scientist that his definition becomes more common than other competing one.
It would be interesting nevertheless to come up with such a list of success criteria for definitions.
I would rather say "lack of specificity" where demonstrability is a subset, that applies to definitions dealing with concepts.
Regards, Evgenij