Given that this distinction is entirely subjective (no-one will be able agree on what a test that pigeonholes pages as a 'site' or 'app'), how is it a useful distinction?

It may not be a useful distinction, but it's useful to have a sense for where on the continuum the thing you're building sits, in order to make technology and architecture decisions. If you're building something more "website-y", it should probably have an architecture that optimizes for search-ability and static caching, whereas if you're building something more "webapp-y", the architecture should probably optimize interactivity and dynamic updates. Most things end up somewhere in the middle, but leaning one way or the other.

I don't think anyone is trying to partition projects into disjunct sets. It's more of a continuum. I think the idea of "app-iness" just captures the interactivity and statefullness of an experience. Incidentally, that additional complexity is exactly what the Web technologies were initially ill-suited for and what frameworks were built to help manage.

Is a "simple website" and a "complex website" a better distinction in your opinion?

Thus I am a complex website engineer. It sounds ridiculous. The webapp developer is domain knowledge wise closer to traditional "application development" than to traditional "website development".

Are sites like vox or gawker considered websites or web apps? Their custom CMS is quite expansive. How bout NYTimes? Is YouTube a web app or a web site? I'd be curious to know some examples on either side. Preferably several closer to the line but on one side or the other rather than far away.

I think you answered your own questions.

I totally agree web site and web app are used interchangeably sometimes -- I was just answering the op as to how I and many people I know classify them and their differences as they relate to the topic of the article, javascript.

For me a web app is an HTML5 web page, that has its interaction logic coded in JS and presentation is done by altering HTML5 DOM and where JS uses REST to send/receive data.

In other words as someone else noted, it is a sandboxed application with

sandbox execution environment == browser IO == REST UI = HTML5 DOM manipulation of the same page

A website (or server app) is a series of HTML5 web apps or HTML5 static pages (JS-less web apps) with navigation logic and inter-app state living in a server app.

Why the emphasis on REST? Websockets tend to be easier for complex applications.

It's not entirely subjective.