How does Flatten really make the coefficients of a vector matrix small in LWE
The important part you missed is the last sentence of the first block:

$BitDecdomp^{-1}$ is still well-defined, if it doesn't get a $N$-dimensional bit vector (all entries from $\{0,1\}$).

In the second quote,  $Flatten$ is defined as first using $BitDecdomp^{-1}$, and then use $BitDecomp$. So the output of $Flatten$ will be a bit-vector. This will be the same as the input only if that input was also a vector from $\{0,1\}$ - but it is defined for any vector over $\mathbb{Z}_q$.

The basic idea here is to use a continuation from a smaller domain to a larger one.