Question about variables in predicates.$$

In this textbook example, the x variable switches positions in O$($x$,$y$)$ in question $9$ and $10.$

I can see that for O$($g$,$x$)$ in question $9,$ x is used to symbolize the dog, as it is used in D$($x$),$ but I was wondering why we can do this?$$

For each sentence are we allowed to change the variables in the symbolization key depending on what we are trying to symbolize? For example, changing O$($x$,$y$)$ to O$($y$,$x$)$ in question $10,$ because we want to use the same variable x that we used in D$($x$)$to represent the ownership of the dog in O$($y$,$x$)?$