There is nothing wrong with what Carroll wrote, which is why it is not in his errata.
Khan is talking about symbolic contracted indices, which must occur in pairs. You sum over their possible values (typically 0, 1, 2, and 3). In Carroll’s equations the indices already have explicit values and are not being contracted. It doesn’t make sense to “contract over 0 and 1” because you can’t assign values to them.
Carroll’s equation follows from a correct double contraction,
when you set $\mu=0$ and $\nu=1$, write out the 16-term double sum over the contracted indices $\alpha$ and $\beta$, and use the fact that all the off-diagonal elements of the Minkowski metric and its inverse vanish. After doing this a few times, you can do it in your head.
It would be instructive to understand why contracting a tensor over a pair of indices produces another tensor, but that is beyond the scope of this particular question.