Can someone tell me the differences between the two types of selections in the pictures below (half square and full square)?
According to KDE Visual Design Group/HIG/CheckBox:
A check box is a control that permits the user to make multiple selections from a number of options. Check boxes are used to toggle an option on or off, or to select or deselect an item. Users make a decision between two clearly opposite choices, e.g. 'on vs. off', 'apply vs. don't apply', 'show vs. hide'.
And further down there's an illustration of the partially filled checkbox with this:
Use the mixed state only to indicate that an option is set for some, but not all, child objects. ...
And this is what I see with a clean, default user account on Kubuntu 18.04:
In the comment section below this answer, user26687 has provided a link to a comment by a developer, d_ed, in There are several settings in KDE that have a "half-checked" state. What do they mean?:
That the current value is evaluated by something else but can be overridden here.
So what you're seeing is by design even if it's not very obvious regarding the function!