Why are there exactly 5 types of two-dimensional lattices, and what distinguishes them?
First, about the black lines. I think they are helpful for the hexagonal and the rhombic lattice. For the hexagonal system, it shows you the hexagons. One hexagon contains exactly the same space as one conventional unit cell. You just have move (translate by a lattice edge) certain areas as shown below.

For the rhombic lattice, it is nice to have the black lines when you are looking at the green cell. Without it, you would not understand why you would call it rhombic.

So why does a rhombic lattice have 2 types of unit cells (the blue and the green) whereas a hexagonal lattice only has 1 (and why do the solid black lines for these two lattices are different?).

The green cell is a centered cell, which makes the math easier because it has right angles. On the other hand, you suddenly have a translation by $½a + ½b$ on top of the lattice translations that you have the keep track of. In fact, the rhombic type is usually described as a centered rectangular lattice,  e.g. https://en.wikipedia.org/wiki/Bravais_lattice#In_2_dimensions.

The same question also applies to rectangular lattice. Isn't it just a special case of oblique lattice where γ = 90°?

Compared to the oblique lattice, all the other lattices are special cases. They aren't special cases because of some rare coincident, but because of the presence of point group symmetries, i.e. mirror planes and twofold, threefold, fourfold and sixfold rotations. A mirror plane will always allow you to choose a unit cell with a right angle. A fourfold axis will always allow you to choose a unit cell that is square, and a threefold or sixfold axis will always allow you to choose a unit cell that is of the hexagonal type. As you can see, the conventional unit cell is not a hexagon but a rhombus with a 60 degree angle. This makes the math easier.
I don't know why the illustrator chose to offset the black lines and the unit cell by half a unit cell length in both $a$ and $b$ for all lattice types (except the hexagonal one). Maybe it is for better visibility. In general, any point in the unit cell will have symmetry mates in other unit cells related by lattice translations.
If you want to see some examples of lattices that contain objects, check out this link: https://www.iucr.org/education/pamphlets/21/supplement
Here is the illustration for a lattice that contains a hexagonal axis. In this case, you have six identical objects in the unit cell, related by point group symmetry. They do no fit directly into the conventional unit cell, but it has the same volume as six objects (you can try to draw the black lines showing hexogons in your mind). The object is called the asymmetric unit, and it can have various shapes as long as it maps onto the unit cell filling the space without overlap.