Also situations where components are connected in parallel and next to each other, I see there isn't really a 'trace' as such run between them, it looks as if one large 'rectangular' area exists with two thru holes. What is the terminology for that?
Many names, depending on the PCB design software used.
Plane - this name is usually used when the area is fully or nearly contiguous, i.e., there are no other tracks that need to be avoided. Just like a copper sheet, with just enough etching to remove copper around those vias that do
not connect to this plane.
Polygon pour or polygon fill - basically the same, but this term is used in situations when there something else (tracks) on the same layer. Polygon pour looks for the empty areas and "flood fills" everything, keeping configured minimum safe distance to existing tracks. The idea is to automatically produce as wide "tracks" as possible. Manually drawing wide tracks leaves a lot of unused space, you see.
Fill, poly region, region, etc. - an arbitrary shape manually drawn by the designer. Basically just a wide "track".
The advantage of wide fills is, you get minimized resistance and inductance, and also better thermal conductivity.
Very usually in 2-layer extra low voltage design, one just draws all the traces
except the ground tracks, and after done, fills every gap with ground polygon pour. If you have more layers at your disposal, then you usually can dedicate some of them as ground or power planes with very little or no other traces running on them at all.