I think you've certainly hit the meaning of your first case. I'd add that sliders are more often horizontal in orientation than vertical, but you can find vertical instances.
"Slider" is also a reference to the control of a button, bar or control-point that moves along a pre-defined path to select an option or value.
Sliders are often used to reduce user effort and input. To select a value, you just need to drag the slider until the required value is met, whereas, a select box/drop-down requires at least double the clicks and a text box infers using the keyboard. They've become a popular way of picking values and reducing user error too.
The second item (a slideshow) is different. These are often called carousels, as they can rotate or cycle images/content without user-input. The example you show has input controls in the shape of both prev/next arrows along with individual content links. The fact that it has been bundled in a gallery of plugins in moot...
Whilst it might cycle through what we know historically as "slides", it's not a slider and I doubt people would understand it as one.