The semistable locus depends on the linearization, not only on the sheaf

Given: AC inherited from the Proj construction (The Axiom of Choice); the one-point projective variety $X={\mathbf{P}}^0=\mathrm{Proj}\mathbb{C}[t]$, the group $G={\mathbf{G}}_m={\mathbb{C}}^{\times }$ acting trivially on $X$, the invertible sheaf $L={\mathcal{O}}_X$ (which is trivial and ample because $X$ is affine), the trivial linearization $L_0$ of $L$, and the linearization $L_{\chi }$ obtained from $L_0$ by the twist by the identity character $\chi :G\to {\mathbb{C}}^{\times }$, $\chi (t)=t$ (G-linearizations of invertible sheaves on a complex G-variety).

The variety and its sections. $X=\mathrm{Proj}\mathbb{C}[t]$ is a single point $p$; ${\mathcal{O}}_X$ is the trivial invertible sheaf, so $\mathrm{\Gamma }(X,L^{\otimes n})=\mathbb{C}$ for every $n\ge 0$ and the global section $1$ is nowhere vanishing (Points of Proj of a graded ring, projective variety classical). The trivial sheaf on a one-point scheme is ample: the nonvanishing locus of the constant section $1$ is the affine scheme $X$ itself (Absolute ampleness by affine section opens).

Linearizations on the point. The total space of ${\mathcal{O}}_X$ is the one-dimensional vector space $\mathbb{C}$, and a linearization of ${\mathcal{O}}_X$ for the trivial action on $X$ is precisely an algebraic group homomorphism $\chi :G\to {\mathbb{C}}^{\times }$ together with the action $t\cdot z=\chi (t)z$ on the fibre; the trivial linearization is $\chi =1$, and the twist of a linearization by a character multiplies the fibre action by that character (G-linearizations of invertible sheaves on a complex G-variety).

Semistable and stable points. A point $x$ is semistable for a linearized ample sheaf if some invariant section of a positive tensor power does not vanish at $x$, and it is stable if in addition its orbit is closed in the semistable locus and its stabilizer is finite; the quotient is $\mathrm{Proj}$ of the invariant section ring (Semistable and stable points for a linearization, The invariant section ring and the projective GIT quotient).

Proj of the invariant rings. $\mathrm{Proj}\mathbb{C}[t]=X$ is the one-point scheme: $D_{+}(t)$ is its whole space and has chart ring $\mathbb{C}[t,t^{-1}{]}_0=\mathbb{C}$ (Standard opens are affine), while $\mathrm{Proj}\mathbb{C}=\mathrm{\varnothing }$ because the ring $\mathbb{C}$ concentrated in degree zero has $S_{+}=0$ contained in every homogeneous prime (Points of Proj of a graded ring). The stabilizer of $p$ is $G$ itself, which is not finite and has dimension $1>0$ (Global and local dimension of classical varieties).