Editing a paper today, I came across a word that I find myself correcting often. One sense of the verb "to coin" is "to frame or invent", but the OED is careful to specify what is normally being invented, namely, "a new word or phrase". One does not "coin" a theory or an idea, nor is the act of naming something itself "coinage". Coining is something one does to terms, and what one does is to create them, usually, the OED tells us, for a "deliberate purpose". The verb is also "occasionally used depreciatively, as if the process were analogous to that of the counterfeiter".
In academic writing, I suggest you stick to using "coin" only when saying something like
Wittgenstein coined the term "language game" to describe an essential part of his philosophical method.
Please note that it is not correct to say "Wittgenstein coined this part of his method a 'language game'". You always coin the term, not the thing it refers to. He would have called it a "language game".
Finally, remember that the phrase "to coin a phrase" is always used ironically, immediately after uttering a cliché for example. That is, one only says "to coin a phrase" when this is obviously not what one is doing. The phrase may have been coined for this deliberate purpose.