special relativity - If $Lambda$ isn't a tensor, what is the meaning of $Lambda ^mu _{~~~nu}$ and $Lambda _{mu nu}$ and so on?

Following this question that asserts that $\Lambda$ (the transformation matrix in Lorentz group) is not a tensor, then if $\Lambda^\mu_{~~~\nu}$ is THE Lorentz transformation matrix, what is the meaning of $\Lambda_{\mu \nu}$, $\Lambda_\mu^{~~~\nu}$ and $\Lambda^{\mu\nu}$ ?

I know how are they related to $\Lambda^\mu_{~~~\nu}$, for example:

$$\Lambda_{\mu\nu} = \eta_{\mu\sigma} \Lambda^\sigma_{~~~\nu}.$$ Considering the fact that $\eta$ is indeed a tensor and $\Lambda$ is "just a number" (well, just a matrix), does this mean that $\Lambda_{\mu\nu}$ is a tensor in the same way that if $\vec{r}$ is an ordinary 3D vector and $a\in\mathbb{R}$ just a number then $\vec{a} = a \vec{r}$ is a vector?