I wonder whether deformations of a vector bundle $F$ over a local Artinian ring $(A, m)$ form a vector space $\operatorname{Ext}^1(F, F) \otimes m$, or what happens in a sequence (*) below? I read two different discussions, one in Huybrechts, Lehn Geometry of moduli space of sheaves, another in Hartshorne Lectures on deformation theory, but it still looks bizarre.
Definitions. Let $F$ be a vector bundle on a nice variety $X$ over a field $k$. Let $A$ be a local Artinian ring over $k$, for example $A=k[t]/(t^n)$, and recall that a deformation of $F$ on $A$ is a vector bundle $F'$ on $X \times \operatorname{Spec} A$ with a fixed isomorphism $F'|_X \to F$. Denote by $D_F(A)$ the set of all deformations of $F$ over $A$, so that $D_F$ is a covariant functor from local Artinian rings to sets. A small extension is a sequence $$0 \to K \to A' \to A \to 0$$ such that the maximal ideal $m'$ of $A'$ goes to the maximal ideal $m$ of $A$, and such that $Km'=0$ holds. A typical example is $$0 \to k \xrightarrow{t^n} k[t]/(t^{n+1}) \to k[t]/(t^n) \to 0.$$
Theorem. Every small extension gives a sequence $$\operatorname{Ext}^1(F, F) \otimes K \to D_F(A') \to D_F(A) \to \operatorname{Ext}^2(F, F) \otimes K,\qquad(*)$$ which is exact in a sense of sets with a chosen element, and also functorial.
But $D_F(k)=0$, so $D_F(A)=\operatorname{Ext}^1(F, F) \otimes m$ holds at least for $A=k[t]/(t^2)$. So does this equality holds for a general $A$? And why a sequence $$\operatorname{Ext}^1(F, F) \otimes K \to \operatorname{Ext}^1(F, F) \otimes m' \to \operatorname{Ext}^1(F, F) \otimes m \to \operatorname{Ext}^2(F, F) \otimes K$$ is so similar to a long exact sequence for $\operatorname{Hom}$-functor? At last, what is a correct way to think about the sequence (*): Huybrechts, Lehn choose an injective resolution, while Hartshorne uses Čech cocycles, but both look like just ways to calculate a cohomology by hand.
Deformations over a general Artinian ring won't be a vector space: they are points of a moduli space (the moduli of vector bundles), which is (typically, and certainly in this case) non-linear.
The deformations over $k[t]/(t^2)$ do form a vector space: this is a general property, namely that the Zariski tangent space at a point of a scheme is a vector space. In the case of deformations of a vector bundle, it admits the description as an $\mathrm{Ext}^1$ that you recall.