Mounting hole on a PCB
The mounting holes that have the large copper pads around the hole are fabricated that way because it is desired to have that connected circuit be conductive to the mounting screw. In most cases it is the GND of the circuit that connects to the mounting hole pad and there is a desire to have such GND connect into the metal chassis to which the board mounts.

The small via holes in the mounting hole pads are designed to electrically connect the mounting hole pad to the pad on the opposite side of the board. In the case of a multi layer board with an inner plane layer the vias may also be used to connect the mounting hole pad to that inner layer. 

In the past it was much less common to see mounting hole pads with these via holes. Instead the mounting hole was built as as a large plated through hole to connect to the opposite side and/or inner layers of the board. However, as in the examples you show, note that the more modern type mounting holes do not have plating in the large diameter screw hole and thus the need for the vias to provide the electrical connection. 

Plating removal in screw mounting holes is primarily done for one reason. The sharp threads of a screw in the hole can cause small particles of metal to come off the plated hole. This is particularly true when boards might be removed and re-installed during testing and/or repair. These small particles can come out and appear on the circuit board or float around in the electronics enclosure. In the advent of surface mount components with vary narrow lead spacings these small metal particles from the screw holes can lead to shorts on the circuit leading to either intermittent circuit operation or outright failure.