Contravariant first index, covariant on second, Vice versa?

[Orodruin]

My understanding is that @Orodruin suggests that there is a problem with what I quoted

No. What you quoted is fine. What you are not paying proper attention to is the difference between MTW’s $L^{\alpha’}_{\phantom\alpha\beta}$ and their $L^{\alpha}_{\phantom\alpha\beta’}$, which are different sets of transformation coefficients. The primes are important in this notation, they tell you what transformation coefficients are intended. You cannot hope to properly recover the transformation rules unambiguously if you drop the primes on the indices unless you introduce different notation for the transformation coefficients.

 

[Orodruin]

For usage, we might transform a vector using the first transformation matrix, i.e $x^{\mu} = \Lambda^{\mu}{}_{\nu} \, x^{\nu}$ and covectors using the second, $x_{\mu} = \Lambda^{\nu}{}_{\mu} x_{\nu}$. Which would be my guess as to what you wanted to do.

Just to be a bit clearer. You are using
$$
x_\mu = \Lambda^\nu_{\phantom\nu\mu} x_\nu,
$$
which is unclear for many reasons. First of all, it is not true taken at face value unless $\Lambda^\nu_{\phantom\nu\mu} = \delta^\nu_\mu$. You need to prime one of the sets of coordinates. Second, if you do this, it is not clear how you intend to handle the indices. If you just leave the indices as they are without priming, you would obtain
$$
x'_\mu = \Lambda^\nu_{\phantom\nu\mu} x_\nu.
$$
There is now nothing that tells me that the $\Lambda^\nu_{\phantom\nu\mu}$ is in any way different from the $\Lambda^\nu_{\phantom\nu\mu}$ that appeared in the contravariant transformation rule. This is the reason for priming the indices belonging to the primed coordinate system. You would end up with
$$
x'_{\mu'} = \Lambda^\nu_{\phantom\nu\mu'} x_\nu
$$
as well as
$$
x'^{\mu'} = \Lambda^{\mu'}_{\phantom\mu\nu} x^\nu,
$$
where the transformation coefficients are clearly distinct.