Do NFAs with $\varepsilon$-moves never terminate?
Sipser proposes an operational intuition for non-determinstic machines. As you have experienced, it's not very useful. In particular, it doesn't relate a lot to the formal definition, and there are more problems down the road¹.

I recommend sticking to the definition: an NFA accepts an input if there is a path from a starting to a final state labelled with the input. That really tells you all you need to know, and all there is to know.

Try to think mathematically instead of computationally.




Parallelism and non-determinism are not the same; you can not build non-deterministic machines in a meaningful sense; cost models (e.g. in complexity theory) for non-deterministic machines don't make sense if you have the split/fork intuition; ...