ag.algebraic geometry - Intuition/Heuristic behind I/I^2 definition of Kähler differentials

Hello,

this one has always been mysterious to me. The Kähler differentials $Omega_{A/k}$ are definined, by the universal property
$$Der_k(A,M)=A-Mod(Omega_{A/k},M)$$
so for $M=A$ we get that $Omega_{A/k}$ is the cotangent space of $spec(A)$.
(or a relative version of it if k is no field).

There are two constructions of Kähler Differentials I know.
The first one is $$Omega_{A/k}=langle df : text{relations satisfied by any derivation} rangle$$
I think I sort of understand this one, it says that the differential of a function just contains enough information to extract the derivation of the function out of it.
And this is what a section into cotangent space should be. Something that contains just enough information to pair it with a vector-field into a function.
The other construction is
$$Omega_{A/k}=I/I^2$$
Where $I$ is the Ideal of functions vanishing on the diagonal in $spec(A)times_{spec(k)} spec(A)$.

More geometrically it says
sections into cotangent space=functions vanishing on the diagonal mod higher order.
But still I don't think I understand this equality on an intuitive level. Can someone explain the heuristic behind this equality? Or maybe explain $Omega_{A/k}=I/I^2$ from another intuitive viewpoint?