Confused b/w non-deterministic finite state automata vs finite state automata
The automaton in the image is indeed nondeterministic.

You need to think a little differently about nondeterministic machines. It's no longer a question of "what happens when X?" but about "what can happen when X?". For each input, there can be many different computations, even with different outcomes. 

Note how the definition of acceptance is different. For DFA, it's "the (unique) computation is accepting"; for NFA, it's "there exists an accepting computation".

Here, in your example, if the input is "twenty", there are two computations:


$q_0 \to q_1$ and
$q_0 \to q_2$.


Since the second is accepting, "twenty" is in the language of this automaton.



Side note: A TCS book that uses twenty as terminal symbol and generally avoids LaTeX for it's mathematics typesetting is highly suspect. I recommend that if you run into any problems, look for a standard textbook before worrying too much.