Excellent question.
Definitions are a little more complicated than you present them. Robinson in his Definitions covers two major categories of definitions: one is the real definition, which he believes shouldn't be a definition at all; the other is a lexical definition which is a linguistic artifact that is part of any language-game. The first might be better understood as an attempt to articulate a concept, and the latter is might be better understood as an attempt of people to use language correctly. The latter notion of definition is easy, because it is merely achieved by the consensus by people about words that serve a particular purpose. For instance, a precising definition in physics will merely articulate words in such a way to avoid ambiguity and prevent or encourage action. Physicists do this by including math and measurement in their definitions so that if someone else is in doubt, they can use numbers to settle the ambiguity or vagueness. This is particularly important in operational definitions, where what is being defined doesn't physically exist.
Of more interest to this question is Robinson's notion of a real definition. It is what is commonly recognized as a discussion of what is real and existent. Obviously, then, the definition is an ontological tool and helps us to perceive and discuss reality, and is an aspect of what Quine calls semantic ascent. From WP:
[Semantic ascent] is the shift from the material mode of language to the formal one. In the formal mode of language we are at a different level. Rather than talking about miles as objects we are talking about what this word 'mile' even means, what it refers to and if it even refers at all. In the formal mode, people with different conceptual schemes might be able to have a reasonable discussion because they are talking about something their conceptual schemes have in common: language.
Thus, we have language, which represents reality, and we have the physical universe which is often considered the foundation of reality. A real definition is means by which come to agreement about reality, which is no easy task, but when successful establishes a type of intersubjective agreement. Educated people in 12th century Rome had one view of reality, and their counterparts in 21st century Chicago have a different view. In fact, the role of definition in this sense is largely about preserving a worldview.
Now, how do we settle on definitions? First, not all people agree on all definitions, either in formulation or method. Language communities agree, but there are many different language communities in the world. How Hitchens, Dawkins, Harris, and Dennett define God, for instance, greatly differs from how the Pope and his Cardinals do. How are we to know who is "right" and who is "wrong"? Perhaps the answer lies in the question 'Do the terms right and wrong or true and false apply to worldviews at all?' Perhaps pragmatic standards such as adequacy and inadequacy are better descriptors.
Whatever your views on the generalization of truth conditions, know that there are three common ways people define things:
- The genus-differential definition - how things are the same; how they differ.
- Definition by sufficiency and necessity - what characteristics are required and/or optional
- Prototypical definitions - how many and of what sort of characteristics does something have
To explain each would be to go beyond the scope of this question. Suffice it to say, that if you master each of these techniques, you'll have a better notion of which strategy is the best strategy in a given context. I say that because definitions are motivated by needs and in different situations. Now we can answer your question:
How do we know that we define something properly when any exception could either mean that the definition itself is bad and needs to be changed to fit better data or the exception to the definition is inconsequential because of some implicit understanding in the word/definition itself?
"Properly" depends on the context, so there is no one right answer. Prototype theory, however, handles just the case you invoke, a case with exceptions or contradictions of extension. This stems from Wittgenstein's reflection on family resemblance. From WP's article on prototype theory:
In this prototype theory, any given concept in any given language has a real world example that best represents this concept. For example: when asked to give an example of the concept furniture, a couch is more frequently cited than, say, a wardrobe. Prototype theory has also been applied in linguistics, as part of the mapping from phonological structure to semantics.
So, if there are exceptions to the rule, or if there is a deviation from the best example, that's fine, because membership to the category is continuous, not discrete. That is, one needn't say a definition is true or false of an object, but that there's a degree of belonging. In other words, rather than expecting a definition to work as a yes-it-is or no-it-isn't, you can also use definitions to say, it is somewhat.
There's a lot to the nuance of definition, and it's not possible to cover the objections and the responses here, but suffice it to say, if you get Robinson's book and give a read, you'll be better off than before. The art of definition is no simple.