That's not how non-determinism works, though perhaps it's how you'd simulate it in real life. Here are several ways of thinking about non-determinism.
The genie. Whenever the machine has a choice, a genie tells it which way to go. If the input is in the language, then the genie can direct the machine in such a way that it eventually accepts. Conversely, if the input is not in the language, whatever the genie tells the machine to do, it will always reject.
Hints. The machine computes a bivariate function. The first input is a word $w$, and the second input is a "hint" $x$. Whenever the machine faces a non-deterministic choice, it consults the next hint symbol, and operates accordingly. 125
We are promised the following:
* Completeness: if $w \in L$ then there is some hint $x$ which causes the machine to accept.
* Soundness: if $w \notin L$ then the machine rejects on all hints.
Accepting computations. An accepting computation is a legal computation (one in which the machine always operates according to one of the choices it is faced with) which ends at an accepting state. A word is in the language iff it has an accepting computation.
We can formalize the notion of accepting computation using snapshots. A snapshot is a triple $(q,z,t)$, where $q$ is the current state, $z$ is the part of the word which remains to be read, and $t$ is the contents of the stack. We can define a relation $(q_1,z_1,t_1) \vdash (q_2,z_2,t_2)$ which expresses that the machine can reach snapshot $(q_2,z_2,t_2)$ from snapshot $(q_1,z_1,t_1)$ in one step. An accepting computation for a word $w$ (for the acceptance condition of emptying the stack) is a sequence $\sigma_0 = (q_0,w,\bot),\sigma_1,\ldots,\sigma_N=(q,\epsilon,\epsilon)$, where $q_0$ is the initial state, $\bot$ is the initial state of the stack, $q$ is an arbitrary state, $\epsilon$ is the empty word/stack, and $\sigma_{i-1} \vdash \sigma_i$ for all $i$.
Another equivalent description is in terms of reachability. Consider a directed graph in which vertices are snapshots and there is an edge from $\sigma$ to $\tau$ if $\sigma \vdash \tau$. An accepting computation is a path from $(q_0,w,\bot)$ to $(q,\epsilon,\epsilon)$, for any state $q$. 399