To be clear, by client-server they mean a connection from a client application directly to the database. For example, in PostgreSQL you can use the command-line tool psql or a graphical tool like pgAdmin. Typically these client applications connect to the database over the network. It's impossible to connect to SQLite over a network, unless you put its database file on a network filesystem (which is risky, as pointed out in the page you link to).
With web apps, the clientele don't connect directly to SQLite. They connect to the web server, like Apache, and through some PHP or Python or whatever, your web server connects to SQLite. From the point of view of the database, your web app is the only customer.
As for multiple writers (which, again, would be going through your web app), the same page has this assurance: "For many situations, this is not a problem. Writers queue up. Each application does its database work quickly and moves on, and no lock lasts for more than a few dozen milliseconds."