Contravariant rank-2 tensor transformation in index notation

I’m slightly confused about the placement of upper and lower indices for the transformation of a rank-2 contravariant tensor.

A contravariant rank-2 tensor transforms as $$M’ = Lambda M Lambda^{T}$$.
In index notation, $Lambda = Lambda^{i}{}_{j}$ and $Lambda^{T} = (Lambda^{T})^{i}{}_{j}$. Therefore, I assumed the above transformation would be written

$$tag{1}(M’)^{ab} = Lambda^{a}{}_{mu}M^{munu}(Lambda^{T})^{nu}{}_{b}$$,

however, it appears in the literature as

$$ tag{2} (M’)^{ab} = Lambda^{a}{}_{mu}M^{munu}(Lambda^{T})_{nu}{}^{b}$$ .

I know that $(Lambda^{T})^{nu}{}_{b}$ and $(Lambda^{T})_{nu}{}^{b}$ differ by a multiplication of two metrics, so these are not the same matrix. However, the only reason I can see to use (2) over (1) is so that a contraction is over an upper and lower index.

Any help in explaining why it’s (2) that we use, not (1), would be greatly appreciated.