[Sharlin]

Hmm, in my experience insofar as anyone really refers to the web these days (mostly tech people), it's the web. Browsing the web, web browser, web server, web app, websockets, WebGL, and so on. Normal people just use "the internet", and have always used, not really aware or caring about what the difference is. Approximately nobody has called the web "world wide web" or "www" in the past 15 to 20 years, and nobody has even said "www" after essentially all URLs went www-less (and good riddance!)

As a sibling commenter said, these days most people don't even really refer to the internet often, because it just is, ubiquitous, ever-present, and accessed via apps that abstract it out even more. Being connected is the default state of matters, and does not require a specific term. Just like you don't often need to refer to or think about the air surrounding you.