As you state, the Euthyphro is a dialogue on the importance of definitions. The question is whether we can actually use words correctly without knowing the precise definition of words. Let me make two points.
First, it is impossible to have precise definitions, because we live in a world of people who are unique. All people see things differently and their experiences are different. In other words, there is no such thing as objective reality. Everything is interpreted and processed by people. This last point makes all knowledge relative.
Second, even though point one is true, communication and understanding is still possible, because definitions, while not perfect, are acceptable. The person who has done the most original and important work on this topic is Wittgenstein. His theory of family resemblances makes Socrates's point invalid. We do not need precision definitions, not to mention this is an impossibility.
Here is what Wittgenstein states:
Consider, for example, the activities that we call “games.” I mean board-games, card-games, ball-games, athletic contests, and so on. What is common to them all? – Don’t say: “They musthave something in common, or they would not be called ‘games;’” – but look and see whether there is anything common to all. – For if you look at them, you won’t see something that is common to all, but similarities, affinities, and a whole series of them at that. To repeat: don’t think, but look! – Look, for example, at board-games, with their various affinities. Now pass to card-games; here you find many correspondences with the first group, but many common features drop out, and others appear. When we pass next to ball-games, much that is common is with noughts and crosses. Or is there always winning and losing, or competition between players? Think of patience. In ball-games, there is winning and losing; but when a child throws his ball at the wall and catches it again, this feature has disappeared. Look at the parts played by skill and luck, and at the difference between skill in chess and skill in tennis. Think now of singing and dancing games; here we have the element of entertainment, but how many other characteristic features have disappeared! And we can go through the many, many other groups of games in the same way, can see how similarities crop up and disappear. And the upshot of these considerations is: we see a complicated network of similarities overlapping and criss-crossing: similarities in the large and in the small."