Two inputs on self loop, deterministic or non-deterministic state machine?

Wikipedia states that a Deterministic State Automation "produces a unique computation (or run) of the automaton for each input string".

I always understood this as there being only 1 possible path to compute any unique string. In which case, the following is a DSM.

But now i'm overthinking this and interpreting the description as each input string having a single possible path, and that path is unique from all other input strings. In which case, the following isn't a DSM as '11' and '12' follow the same paths.

So my question is, is the following a DSM or NDSM?

Its still deterministic, there is only one possible path for each input from each state. 1, and 2 can only go back to itself, for it to be non deterministic, the input should have multiple possible paths. Such as if input 1 had two possible states branching from one specific state.

In short, if there are not branching paths for a specific input, and no ε-edges in the graph, it should be deterministic. i.e. no branching paths, we can determine for sure where it goes. The one you drew above we can always determine the path for a specific input.