We all use language to communicate an argument or a critique of an argument, but we don't pay enough attention to the language we use, the definitions we assume, and what we are really communicating. No matter what the topic being debated, you'll do better if you stop to reflect on the language and meanings being used. That, in turn, is best achieved through some familiarity with the nature of language and communication, the nature of definitions, and how they relate to logical arguments.
Although it might sound trivial or even irrelevant to bring up such basic matters as language, meaning, and communication, these are the most fundamental components of arguments - even more fundamental than propositions, inferences, and conclusions. We cannot make sense of an argument without being able make sense of the language, meaning, and purpose of what is being communicated in the first place.
Logic is the study of methods of reasoning and argumentation, both proper and improper. The Philosophy of Language involves the study of how our language interacts with our thinking. Atheists and theists don't typically disagree much in these subject areas, but they are nevertheless critical because a proper understanding of them is necessary for the sake of sound reasoning, rational thinking, and reliable communication.
It may be true that defining key terms and concepts is important to the start of any productive argument, but not all definitions are the same. Most logic texts list five different types of definitions which people might use. This can give the impression that the matter is pretty well settled, but it's not. In fact, the assumption that the matter is settled is part of what gives rise to debates about the nature of definitions every so often. It might seem strange that a lot of ink and time is spent on something so basic a a "definition," but sometimes it is the most basic of concepts that are the most problematic.
Understanding the difference between
denotation and connotation is important to understanding definitions and how concepts are used. Unfortunately, that is complicated by the fact that these terms can be used in two different ways: grammatical and logical. Even worse, both uses are worth keeping in mind and both uses are relevant to project of logical, critical thinking.