= Solution
Let $\mathfrak m$ be the maximal ideal of the <local ring> $A$. Every open subset of $\operatorname{Spec}A$ containing the <closed point> $\mathfrak m$ is the whole <spectrum of a commutative ring>: it contains a <principal open subscheme> $D(a)$ with $a\notin\mathfrak m$, and that $a$ is a <unit>, so $D(a)=\operatorname{Spec}A$.
Given $f:\operatorname{Spec}A\to\mathbb P^n_{\mathbb Z}$, choose a standard <affine open subscheme> $D_+(x_i)$ containing $f(\mathfrak m)$. Its preimage is consequently all of $\operatorname{Spec}A$. The <affine-target adjunction for schemes> expresses $f$ in this chart by elements $b_j\in A$ for $j\ne i$, the images of $x_j/x_i$. It is represented by <homogeneous coordinates> with $a_i=1$ and $a_j=b_j$.
Conversely, a tuple $(a_0,\ldots,a_n)$ with some $a_i\in A^\times$ defines a <morphism of schemes> into $D_+(x_i)$ by $x_j/x_i\mapsto a_j/a_i$. Choosing another unit entry gives the same <morphism of schemes>, since the usual <projective space> transition functions identify the ratios. Multiplying all entries by one <unit> does not change any ratio. If two such tuples define the same <morphism of schemes>, choose a unit entry $a_i$ in the first and a unit entry $a_j'$ in the second. In the second chart, the function $x_i/x_j$ pulls back to $a_i'/a_j'$. Because the whole map lies in $D_+(x_i)$, this ratio is a <unit>, so $a_i'$ is a unit too. Equality in this chart gives $a_j/a_i=a_j'/a_i'$ for every $j$, hence $a_j'=(a_i'/a_i)a_j$.
\b[Thus the correspondence is exactly]
$$
\boxed{\operatorname{Hom}(\operatorname{Spec}A,\mathbb P^n_{\mathbb Z})
=\{(a_i):\text{some }a_i\in A^\times\}/A^\times.}
$$
This is the <projective coordinates over a local ring> description.
For a general <ring>, the key open-neighbourhood argument fails. Even a tuple generating the unit ideal need not have any unit entry. For example, over $A=k\times k$ the pair $((1,0),(0,1))$ defines a map to $\mathbb P^1$ whose two points have images $[1:0]$ and $[0:1]$. Neither coordinate is a unit, and no common unit multiple changes that fact. The map lies in no single standard chart. More generally, maps into <projective space> correspond to <invertible sheaf> quotients of $A^{n+1}$; the quotient need not be a free rank-one module outside the <local ring> case.
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