Is the charger included with an Apple Watch purchased from the Apple website?

nop only the puck charger

these days it would have an usb-c connector so you need to get an ac charger with an usb-c connector

or use a usb-c to usb-a adaptor Amazon.com : usb-a to usb-c adapter

their tech spec always list what is in the box for their products

Apple Watch Series 10 - Technical Specifications - Apple

If you go through the ordering process at the online Apple Store, you will find that Apple provides a "What's in the Box" section for each Apple Watch that you are considering purchasing.

Here is an example for the new Series 10 watches:

What's included is the magnetic fast charger, but you will still need a separate power adapter to supply power to that charger. This included charger has a USB-C cable and you will need a respective USB-C power adapter to work with it.