Let $X$ be a $K$-scheme. The sheaf of relative differentials of $X$ over $K$ is the sheaf $\Omega_{X/K}$ such that restricted to an open affine subset $U=Spec(A)$ satisfies: $$ \Omega_{X/K}|_U\simeq (\Omega_{A/K})^\sim. $$
Now, if I am not wrong, it is the cotangent sheaf. Hence, it should be the sheaf of sections of the line bundle given by: $$ L=\{(p,\alpha):p\in X, \alpha\in\frac{\mathfrak{m}_p}{\mathfrak{m}_p^2}\}. $$
Are these definitions indeed equivalent?