Can somebody confirm if the Christoffel symbols of the first kind are tensors? I thought that it was the case, but I've found the notes of Stefan Waner were he assures that they are not
It's kind of a matter of definition. It has been traditional to say that the Christoffel symbols are not tensors, because they don't transform like tensors. If you're of the Wald school, however, you will say that the Christoffel symbols are tensors; it's just that there is a different Christoffel tensor for each coordinate system.
In more detail: the Christoffel symbols are defined as the difference between the Levi-Civita connection and a fiducial connection given by the partial derivatives in some coordinate system, ∇ = ∂ + Γ. The difference of two connections is a connection. However, the fiducial connection ∂ depends on the coordinate system (since it's the partial derivatives of some coordinates), so the Christoffel symbols do too. For this reason, they don't obey the ordinary transformation law for tensors, because that assumes you are transforming the components of a single tensor field; with Christoffel symbols, if you change the coordinates, you also change which tensor field you're working with.