Generally, for a prime $p$, Prüfer $p$ group is defined to be direct limit of the system of groups $\{\mathbb{Z}/p^n\mathbb{Z}\colon n\in \mathbb{N}\}$ with the homomorphisms $\mathbb{Z}/p^n\mathbb{Z} \rightarrow \mathbb{Z}/p^{n+1}\mathbb{Z}$ induced by multiplication by $p$. But on Wikipedia, it is defined with some "characterization":
It is the unique infinite $p$ group, in which every element has $p$ $p$'th roots.
It is the unique infinite $p$ group which is locally cyclic (every finite se of elements of group generates a cyclic group.)
How do we prove the "uniqueness" in these characterizations using the definition which is given in terms of direct limits?