Turing Machine notation
The function $\mathsf{int}$ actually doesn't change a binary number to its base 10 form. Rather, it takes a string encoding some number $n$ in binary, and outputs $n$. When we write the output, we use decimal (base 10), but the output is just a number. It's us who use base 10.

The language $L$ consists of all strings $\alpha\#\beta$, where $\alpha,\beta \in \{0,1\}^*$, such that


$\alpha$ encodes (in binary) an integer $a$ in the range $1$ to $|\beta|$, where $|\beta|$ is the length of $\beta$.
The $a$th bit of $\beta$ is $1$.


It's difficult to say what the $a$th bit of $\beta$ is, since there are several possible conventions:


The "first" bit could be bit number $0$ or bit number $1$; in this case it seems like the first bit should be bit number $1$.
We could be counting bits from the left or from the right; the usual convention is from the left, though if you think as a string as encoding a number, it is more natural to count from the right.


Assuming we are counting bits from the left and starting with $1$, $L$ consists of strings $\alpha\#b_1b_2\cdots b_n$ where $\alpha$ encodes a number $a$ in the range $1 \leq a \leq n$ and $b_a = 1$. This interpretation also matches your examples.