For the most part, this categorization of the era was done after the fact by scholars. For instance, when I saw Gordon Wood speak at the James Madison Program at Princeton, he noted, the Founders didn't self consciously say to themselves, (I'm paraphrasing Wood) "We are living in the age of Enlightenment." (He noted this when explaining the context of why America's Founders were the product of "Enlightenment.")
Yet they did commonly use enlightenment terminology. Terms and qualifiers like "benign" "benevolent," "sober," "rational," "reasonable," "liberal," "enlightened," to name of few. [I blogged about that here; also see Tom Van Dyke's summary of Philip Hamburger's article on a related matter.]
Read for yourself the search engine results when one puts the term "enlightened" into George Washington's extant corpus.
Or this particular quotation of Washington's speaking to the Swedenborgians:
We have abundant reason to rejoice that in this Land the light of truth and reason has triumphed over the power of bigotry and superstition, and that every person may here worship God according to the dictates of his own heart. In this enlightened Age and in this Land of equal liberty it is our boast, that a man's religious tenets will not forfeit the protection of the Laws, nor deprive him of the right of attaining and holding the highest Offices that are known in the United States. [Bold mine.]
The first thinker to self consciously understand theirs was an "age of Enlightenment" -- I have found was Immanuel Kant. That is, apparently, where the meme originated.