Don't have any video examples, but I might be able to clear it up conceptually for you.
First, the difference between swing and sway: Swing is a description of a particular range of motion, whereas sway describes an angle. Or put another way, swing is a movement, while sway is a position.
Here is how I would define swing: "The free release of energy through an arc". For something to be said to be "swinging", it needs to travel through an arc, and it must have a released energy. If you understand that, you can grasp the concept of the various types of swing -- some which result in sway, and some which do not.
Swing can apply to literally any object -- For example, you can swing a bat, a club, or a rope. In dancing, it can apply to any particular body part -- you can swing your arm, leg, or even your head. Or you can swing your body as a whole.
When swing is applied to a specific body part, it is usually labeled as such. For example, we often talk about "leg swing", which is the free release of energy, causing the leg to "swing" through an arc.
Sometimes you'll hear about "rotary swing", "pendular swing", and "metronomic swing". These phrases are used to describe the range of motion, or the plane on which the arc resides. Leg swing, for example, is by nature pendular -- It swings from a fixed point above, like a pendulum. When you go bowling, you also use a pendular swinging action with your arm. Toss a baseball, and the swing of the arm would be described as "metronomic", meaning that it swings from a fixed point below, like a metronome.
Rotary swing is the other common range of motion used in dancing. If you were to swing a baseball bat, that would be taken on the rotary plane. This is a good example of a type of swing, when applied to the body, that does not result in sway on its own (unless you combine it with other ranges of motion). So for example, when you dance a Natural Spin Turn, you are producing a rotary swing, but no sway results.
In ballroom dancing, most of the time we combine these types of swing. For example, on your basic Natural Turn, you have both a rotary swing *and* a pendular swing, causing you to rotate and to sway.
Regards,
Jonathan