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The convention for notating indices of a tensor is to write a contravariant index superscript and a covariant index subscript. If one has a pure contravariant or a pure covariant tensor of $2$nd order, then the association of the $i$th index with the $i$th dimension of the tensor is clear: $$F^{\alpha\beta},\quad F_{\alpha\beta}.$$ In this case, $\alpha$ gives the index of the $1$st dimension, $\beta$ the index of the $2$nd dimension.

However, if it comes to a mixed tensor of $2$nd order, I frequently come across the notation $$F^\alpha_\beta,$$ where both indices are positioned right above each other, directly after the tensor symbol. In my understanding, this neglects the index position and with that the association of an index with its dimension. It is not clear if this notation is intended to mean $${F^\alpha}_\beta\quad\text{or}\quad{F_\beta}^\alpha.$$ Am I missing something?

Even if $F$ was symmetric in the indices $\alpha$ and $\beta$, ${F^\alpha}_\beta\neq{F_\beta}^\alpha$ in general since they transform differently under a transformation $T$: $${\overline{F}^\alpha}_\beta=\left(T^{-1}\right)_{\alpha\mu}T_{\nu\beta}{F^\mu}_\nu\quad\Leftrightarrow\quad\overline{F}=T^{-1}FT\quad\quad\;\\ {\overline{F}_\beta}^\alpha=T_{\mu\beta}\left(T^{-1}\right)_{\alpha\nu}{F_\mu}^\nu\quad\Leftrightarrow\quad\overline{F}=T^\text{T}F\left(T^{-1}\right)^\text{T}$$

Even common literature uses this position-insensitive notation (Theoretical Physics 4 by Wolfgang Nolting, e.g.), and so do some of my professors in particle physics, where contravariant and covariant tensors of $2$nd order appear on a daily basis.

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You're not missing anything at all -- it's simply sloppy notation, and the people who do it just don't want to bother putting in the spacing correctly.

However, you are missing something about the case of symmetric tensors. In this case, there is no ambiguity: an upper index transforms by contracting against the lower index of $\Lambda^{\mu'}_{\ \ \nu}$, while a lower index transforms by contracting against the upper index of $\Lambda^{\mu}_{\ \ \nu'}$.

You might think that it makes a difference if you want to write the contraction as a matrix multiplication. But matrix multiplication is nothing more than a trick for remembering the general rules I just said, and a rather limited one at that. It might be true that the matrix multiplication representation differs between the two cases you gave, but that just means it's adding unnecessary complication. The transformation rule, in index notation, is the real definition, and unambiguous.

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