Re: Distinctions
A perfectly sensible term, since it refers to an aesthetic and cultural movement called "Modernism" (a proper noun), which takes its name from the concept of "modernism", which, contra Chesterton, is itself a perfectly sensible idea.
Modernism as a concept refers to a conscious recognition of living in a world which is significantly different from its recent historical past. People do not always feel this way; most Europeans did not during the middle ages, for example. There are reams of cultural history documenting and analyzing this shift in social perceptions.
Modernism as an aesthetic movement refers to a number of schools and artists who took it as their mission to represent, explore, and/or encourage ideas and techniques they felt were new, or at any rate a break from the past. (It's not to be confused with the artistic avant garde, though, which in effect sought to take a step further, though that's a very simplistic version of the difference.)
Post-modernism refers to a collection of movements in reaction to Modernism.
The Renaissance didn't involve everyone giving birth, either. Names are not bound to exactly represent their etymologies. Chesterton's observation is a sophomorism.
A good reference on the concepts of modernism, Modernism, and particularly the period often called "High Modernism" is Eysteinsson, The Concept of Modernism. There are any number of others, of course. For Post-modernism, the Routledge book of the same title (I forget the author and can't be bothered to look it up) is pretty good. The term was popularized by various architects and by essays such as Jameson's "Post-Modernism, or the Logic of Late Capitalism", but I wouldn't go back to those sources; the term drifted too much afterward.