Projective bundle of a trivial module

The symmetric algebra. Since ${\mathcal{O}}_S^r$ is free of rank $r$, [F1] identifies $\mathrm{Sym}({\mathcal{O}}_S^r)$ with the graded ${\mathcal{O}}_S$-algebra ${\mathcal{O}}_S[T_1,\dots ,T_r]$ with $\mathrm{deg}{T}_i=1$, generated by its degree-one part; hence ${\mathbb{P}}_S({\mathcal{O}}_S^r)={\mathrm{Proj}}_S{\mathcal{O}}_S[T_1,\dots ,T_r]$.

The case $r=0$. The zero module $E=0$ has $\mathrm{Sym}(0)={\mathcal{O}}_S$ concentrated in degree $0$ and ${\mathbb{P}}_S(0)=\mathrm{\varnothing }$ by [F2]; consistently, for every $S$-scheme $T$ a surjection $0\to L$ onto an invertible sheaf exists only when $T=\mathrm{\varnothing }$, since an invertible sheaf on a nonempty scheme is nonzero, and morphisms $T\to \mathrm{\varnothing }$ likewise exist only for $T=\mathrm{\varnothing }$.

The isomorphism with projective space. Applying [F2] with $U=S$, where ${\mathcal{O}}_S^r{\mathrm{\mid }}_S={\mathcal{O}}_S^r$ is free of rank $r\ge 1$, gives ${\mathbb{P}}_S({\mathcal{O}}_S^r)\cong {\mathbb{P}}_S^{r-1}$; under this isomorphism the twist $\mathcal{O}(1)$ corresponds to the standard twist and the tautological quotient restricts to the standard quotient ${\mathcal{O}}_S^r\to {\mathcal{O}}_{{\mathbb{P}}_S^{r-1}}(1)$ whose components are the coordinate sections $x_0,\dots ,x_{r-1}$. This is the isomorphism of (1), and it is canonical because it is the chart-gluing identification of the two constructions.

The case $r=1$. Here ${\mathbb{P}}_S({\mathcal{O}}_S)\cong {\mathbb{P}}_S^0=S$ by step 2.1 and [F2], and ${\mathbb{P}}_S^0$ has the single chart $U_0=S$ with frame $e_0$, so $\mathcal{O}(1)\cong {\mathcal{O}}_S$ with frame the coordinate section $x_0$; the tautological quotient ${\mathcal{O}}_S={\mathrm{Sym}}^1({\mathcal{O}}_S)\to \mathcal{O}(1)$ sends the generator to the coordinate section, which is the frame $e_0$, hence is an isomorphism ${\mathcal{O}}_S\to \mathcal{O}(1)\cong {\mathcal{O}}_S$: under the identification by the frame it is the identity. Equivalently, by [F3] the right side for $E={\mathcal{O}}_S$ consists of isomorphism classes of surjections $g^{\ast }{\mathcal{O}}_S={\mathcal{O}}_T\to L$ with $L$ invertible, and every such surjection is an isomorphism by [F4], so there is exactly one class; correspondingly $T\to {\mathbb{P}}_S({\mathcal{O}}_S)$ is the single structure morphism $T\to S$, and the two descriptions agree.

Conclusion. Steps 1.1 and 1.2 give the isomorphism ${\mathbb{P}}_S({\mathcal{O}}_S^r)\cong {\mathbb{P}}_S^{r-1}$ for $r\ge 1$ together with the identification of the universal quotients, step 3.1 computes the case $r=1$ as ${\mathbb{P}}_S({\mathcal{O}}_S)=S$ with tautological quotient the identity ${\mathcal{O}}_S\to {\mathcal{O}}_S$, and step 1.2 records ${\mathbb{P}}_S(0)=\mathrm{\varnothing }$. The identification of universal quotients is what makes the isomorphism an isomorphism "in the quotient convention" of [F3]: for $E={\mathcal{O}}_S^r$ the functor $T\mapsto \{S\text{-morphisms }T\to {\mathbb{P}}_S^{r-1}\}$ is the functor of surjections ${\mathcal{O}}_T^r\to L$ in both models. The Axiom of Choice [A1] is inherited from the relative Proj construction; no further choice is made. [A1, F3, step 2.1, step 3.1, cases: r=0 and r=1 and r at least 2] \qed