I have troubles with defining a tensor of rank $2$.
As far as I can see a vector has $3$ components and is always a tensor of rank $1$ (one directions). a dyad has $9$ components and is a tensor of rank $2$ (two directions) enter image description here
How can a rank-$2$ tensor has $9$ components even though it has only two directions, which should be resulting in $6$ components.
A tensor of rank $2$ is just a matrix. If your underlying vector space is of dimension $3$, as you seem to assume, then this would be a $3\times 3$-matrix, which obviously has $9$ components. A general tensor of rank $r$ in a $d$-dimensional vector space has $d^r$ components.
The image you linked to does seem to represent a tensor of rank $2$. I wouldn't call this “two directions”, though, as you essentially get three directions, illustrated as $T(e_i)$, the images of the unit vectors. So the tensor in this illustration is used as a transformation matrix. But the actual geometric interpretation of any given tensor may depend on the application.
You might be mixing up the concept of a tensor with that of a bivector. At least to me, the intuitive description of “like a vector, but with two directions instead of one” sounds pretty much like a bivector. But a bivector in three dimensions is very much like a cross product: you get the normal direction of the plane spanned by the two constituent vectors, and some magnitude, but taken together these still form only three components.