Excerpts from your document:
Devices
Home Assistant calls the smart devices in your home Devices. An example would be a Hue lightbulb, an automated garage door opener, or a motion sensor.
Entities
Instead of automating devices, people in Home Assistant usually talk about editing entities. An entity represents some function of a device, like a single sensor or a single switch.
Two definitions, each defining a different concept, yet both use the same example: a sensor is a device and an entity.
Shimmer, it’s a floor wax and a dessert topping. 
You didn’t make a mistake; you identified one of the aspects of Home Assistant’s evolving terminology.
In the beginning, there were only entities. A light bulb is a physical “thing” and is modeled as an entity (whose domain is light). If you have a physical “thing” that reports several properties like temperature, humidity, and light level, it is modeled as three separate entities (whose domain is sensor). However, there’s nothing to refer to the “thing” itself. As a result, there’s no means of showing the three sensor entities belong to a single physical “thing”.
The concept of devices was introduced as a means of identifying a “thing” and all the entities that are part of it. You have a multi-sensor device with three sensor entities reporting temperature, humidity, and light level. The same concept also paved the way for device automations.
For example, here’s an ecobee switch+ device with several entities. I can easily create device automations for it from its Device view:
If I look at the Philips Hue bridge device, it shows no entities.
Each Philips Hue light bulb is shown as a separate device. If you examine the device, it shows it has one entity, a light.
So in this example, a Philips Hue light bulb is both a device and an entity.
Summary
- A device represents a physical thing.
- A device may have zero or more entities.
- An entity is typically a property of a physical thing but can also represent the thing itself.
