Given the open formula: $\alpha =(\exists{{x}_{2}})({P}^{1}({x}_{1},{x}_{2}))$
And consider the interpretation $I$ where the domain is the natural numbers, and ${P}^{1}$ means equality.
Is $\alpha$ true, false, or indeterminate under $I$?
I think is true, because in any possible valuation of the free variable ${x}_{1}$, there exists ${x}_{2}$ such that ${x}_{1}={x}_{2}$. However my "official solution" says it is indeterminate.
Is my reasoning correct?.
The question is "tricky"...
According to some textbook, an open formula (a formula with free variables) is true (in a structure) iff its universal closure is; see e.g. Dirk van Dalen, Logic and Structure (5th ed - 2013), page 67.
According to this definition, you are right: $\mathbb N \vDash \alpha$. In general, with $P^1$ interpreted as equality, the formula is true in every structure.
But there are other approaches; see e.g. Herbert Enderton, A Mathematical Introduction to Logic (2nd ed - 2001), page 84.
According to this approach, the interpretation of a formula $\varphi$ in a structure $\mathfrak A$ is defined with a variable assignment function $s : Var \to |\mathfrak A|$, where $|\mathfrak A|$ is the domain of $\mathfrak A$.
Thus, the basic semantic relation is:
According to this definition, $\mathbb N \vDash \alpha$ is not defined, because we have not specified $s$.
Conclusion: you have to check the details of the semantic specifications of your textbook.