NFA designing for strings starting with $01$ The question was asked

Construct an NFA with set of all strings that start with $10$.

The solution provided to me is

But my question is what if the automaton receives an input $0$ at the starting? Also there is no option for $q_1$ to transit after receiving $1$. So I think the solution should be

Please correct me if I am wrong.
Since this is an NFA, the solutions are equivalent. When you are in a state without any transition for the current input you receive, you terminate (equivalent to getting stuck in a non-accepting state without going out of it).