Turing machine that can tell if it ends in standard position?
In most of the models of Turing Machines there is a separate blank symbol $B$. Sometimes we may even have a start symbol $\rhd$. This makes it easy to check if $M$ halts in standard position.

If $0$ is also treated as the blank symbol, and the tape is initialized with 0's on both side of inputs (with no separate blank symbol) then the analysis of computation becomes a little bit difficult (Where is the boundary?).

In your case, it is doable without much hassle as 0's in the input always occur singly. First rewrite the whole input as 2-bit alphabet, converting 0 to 10 and 1 to 11, B will be 00 and start symbol $\rhd$ will be 01. Now we are happy to go. We just need to modify the Turing Machine to consider 2 bits at a time.