More than one NFA accepting a given language
There are many, many different NFAs accepting a particular language, or DFAs for that matter. You can just add branches that lead into dead ends at will, for example. Or add detours that give the same result as the main road, like in your automaton adding a new state $r$, with transitions from $p$ to $r$ on 1, and then from $r$ to $r$ on 0 and 1, and finally from $r$ to $q$ on 1.

In the given case, omitting the transition from $p$ to $p$ on 0 gives another language. You should convince yourself that the modified NFA doesn't accept anything containing zeros.