makes it sound as if the video/content was streamed from the phone to the Chromecast, which isn't true.

You can stream content directly from your phone or computer. Works great with apps like VLC or command line applications like castnow. On computers and Android you can also mirror the screen and view content that way, so yes, you can stream content directly from a device.

Chromecast is basically a web browser that loads special websites and is remote controlled by a phone.

That is a technical implementation detail that has almost no relevance for the user, especially a new user. In virtually any traditional wired playback scenario you have a playback device, a cable, and a TV. Chromecast takes the place of the cable in this equation. It is essentially inert (like a cable) without a playback device telling it what to play.

Tapping the cast icon in an app sends a link to the Chromecast to a special websites which it loads. It remains a lightweight connection to the phone to control playback, volume or stop casting.

It does a lot more than just send a URL to a special site. A lot of how the Chromecast works is related to how the session between the sender and receiver is established, and then how the Chromecast uses CEC to set up the TV. The Chromecast can use mDNS for establishing a session, or even ultrasonic frequencies (for connecting "Nearby devices"). Even though HTML / CSS / JS can be used to create a custom player, it isn't required for playing back media.