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(That was me previously)
Yes, the subject of the post would be more properly formulated as "What is the significance of definitions?", as definitions in dictionaries obviously do exist.

To answer your question, "then what is primary?", in my opinion understanding from examples is primary.

You say that people need definitions in order to argue formally. You are right, but people don't argue formally most of the time. You don't go to a furniture shop and start arguing whether a certain piece of furniture is a chair. There are areas where formal arguments are important, such as law and mathematics, and in these areas definitions are primary. This means, for example, that a term means precisely what it is defined to mean. One can probably invent a new kind of furniture that people will call a new kind of chair, but one can't commit a new kind of crime that will be called murder though it doesn't answer the legal definition thereof.

In other sciences terms don't have such good definitions. For example, energy is often defined as the ability to do work, but that's not what energy really is. Or take biology. The definition of a taxonomical term is just the most convenient way to distinguish a set of genetically and evolutionarily related species from others. It's the most conspicuous common trait(s). For example, a mammal is an animal that lactates. However, if someone finds a lactating mutant bug, it won't be considered a mammal. Conversely, a non-lactating mutant goat will still be a mammal.

To sum up, a concept is more like a fuzzy cloud of traits, and a definition is like a small well-defined list of the most conspicuous ones.