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.
To get the same holes in your eagle project go to your schematic and click on add (or type add in the text box and hit enter). Now to go the holes library, then you see there is a MOUNT-PAD-ROUND and a MOUNT-PAD-SQUARE library. These are the library's you are looking for. In the schematic you can connect the pad to a signal, which is mostly used to connect to ground (not sure about this). And yes on the blue PCB's they are also functioning as via's probably because the screws are connected to ground.
The holes in your mounting hole is probably there so more current can flow through. To get these holes also in your eagle project you probably need to make your own library for that. The golden finish on the mounting hole is there because it is considered as a via or regular hole.
Another reason for the vias around the mounting hole is wave soldering. When wave soldering solder covers all unmasked copper and therefore the whole mounting hole would be covered with solder. Manufacturers usually mask those holes with some tape manually which costs money. This way you have the electrical connection thanks to those vias that can be soldered through and the mounting hole is preserved and manufacturer didn't have to do extra work.