Four layer board manufacturing question
Though you already got your answer:

Better 2-layer PCBs are made of fiberglass mats soaked with resin. The resin is hardened, the whole thing is plated with copper, and you have your raw PCB.
During production of an actual board, it is drilled two times: First for all vias an through-hole component pads. They are then chemically plated with metal to make them conductive. Second, at some time after the plating process to make mounting holes etc  without plating.

A 4-layer PCB is actually two 2-layer PCBs (cores) glued together by an other layer of soaked fiberglass (prepreg), which is hardened after. This is what your pictured show: Two green cores with the prepreg inbetween.

It is easily possible to create vias from one side of a core to the other, as this is the same as for 2-layer PCBs. But it is costy, since it is an extra step in production. But it is very difficult to create vias through a core and a prepeg, bu not the next core, as the second via style in your image shows. One had to drill to a certain depth into the board, and plating a blind hole is difficult to impossible.

Finally, it is much cheaper to have normal vias going through the entire board, and many manufacturers, especially pooling services, only offer these vias.

There are also very few manufacturers which etch a center core, apply a prepeg on both sides, plate it with copper and then etch the outer layers. And more-than-4-layer boards are made alike. This has different constraints for vias, but you got the point.

For designing in EAGLE:

A ground plane is a polygon typically with name GND. Place a via, and the ground plane will get a cut-out to not touch the via. Name the via GND, and the ground plane will connect to it. During routing of a signal which is already named GND, any of its vias are automatically connected to the GND plane.