I disagree. I don't care if nginx is in front of apache, varnish, or some ruby or python web server. If nginx is what serves the final request to the client, it's the webserver. Period.

edit: nginx is the last mile. You can't say everyone in the USA has access to fiber because further upstream their traffic traverses a fiber network. Their last mile is DSL/Cable/etc. When it comes to the user experience, last mile is all that counts.

If you start calling things like varnish a web server you'll confuse a lot of people. You generally call them according to what role they serve in your stack, and for Varnish and at times Nginx, that is a reverse caching proxy.

By that logic (I'm not saying the conclusion is wrong), no one has fiber because we always end up translating it to electrical pulses before hitting the cpu, and chrome is the largest webserver because it always checks its own cache before going out to the network.

Sure. But it doesn't make sense to not count the other web servers involved. (Even though that's often not possible)

Depends, but from an application and development perspective, which I'm sure most are assuming here, no. If you can remove nginx and still serve your site/application, then nginx is not your web server. If nginx is acting as a reverse proxy or a cache, then that is all it is.

By that logic, the webserver would be the last thing in front of the user, their router, their ISP, whatever...

I dont agree. Their router processes data below the application layer. Thus why not define the webserver as the last server to serve at the http layer ?