Modern should be used to refer to the present time, as opposed to the past. The ancient Romans referred to their society as "modernus" in contrast to the ancient Greeks, Egyptians, and their own ancestors. Medieval people did likewise. Nowadays historians tend to refer to the modern period as starting about five or six hundred years ago, and generally use the term "Early Modern Period" rather than "Renaissance," unless speaking in the context of the widespread artistic and intellectual movements.

This is an interesting question and I don't suppose you will get a single or completely satisfying answer. More sensitive scholars will distinguish between modern, modernity, modernization, and modernism. I would argue that much of the discipline understands "modern" to be referring to a period of time rather than the present. "Modern" is a frame of mind rather than a time period, so you can have "modern" thinkers today, but the modern period is the time when the idea of modernity was most widespread.

The usage of modern seems to apply to a time period in which society at large had an attitude that believed in the possibility of improving the human condition through the application of "reason". It is essentially optimistic, believes in progress and mostly materialistic. While some scholars argue it is also western, I don't think this is necessarily the case, and the scholarship recognizes the existence of "multiple modernities". What unites all modernities are two main tenets: Humans can end the control of nature over humans, and the control of humans by other humans. You can call this emanciption from natural and social exploitation. We first see the rise of this way of thinking in the late 17th century in some parts of Europe (lets say 1680 is early modern) and it catches on in other parts of the world at different times and in different ways. Though to be fair a "modern" attitude can be found at various points in history, including among early Muslim philosophers (eg, Ibn Tufayl). It seems to have tapered off after the 1970s, with most social scientists and perhaps society at large accepting the flaws in the optimistic understanding of human "development" and an appreciation that we may not be endlessly moving forward. Ofcourse other historians argue that we are still in the modern period and that the loss of faith in modernity is insulated and exaggerated.

I guess to answer your question of how it is used, books are still being published and getting away with using the periodization of "early modern". The period of the "modern" is more controversial, but you can also still get away with using it. When does the early modern turn into full blown modern is a topic of much debate and will depend whether you are looking at politics, technology, ideology, gender or another area.

Sources: Bruno Latour: We Have Never Been Modern

Stephen Toulmin, Cosmopolis: The Hidden Agenda of Modernity

William Conolly, Political Theory and Modernity

Francis Bacon and Condorcet are some earlier "modern" thinkers and have accessible manifestos on modernity particularly Condorcet's "Sketch for a Historical Picture of the Progress of the Human Mind"

Edit: The 1680 date isn't a clear cut off point. Many books on Early Modern Europe start at the end of the 15th century. It's more for general reference.

Follow up: does "post-modern" have any meaning in a historical context, or is it strictly an artistic movement?