What is the correct elliptic curve representation?
In general case, each elliptic curve over field $K$ can be defined by the following equation:
$$y^2+a_1 xy+a_3 y=x^3+a_2 x^2+a_4 x+a_6$$
If the characteristic of field be larger than $3$, then all elliptic curves over it are isomorphic to
$$y^2=x^3+ax+b$$
In the case that characteristic is $3$ or $2$, we have two form of isomorphic curves which are as follows respectively.
$$y^2=x^3+ax+b$$
$$y^2=x^3+ax^2+b$$
$$y^2+cy=x^3+ax+b$$
$$y^2+xy=x^3+ax^2+b$$
Edit: The representation of elliptic curve depends on your goal and the field.  Elliptic curves are not defined only on ordinary curves. As an example, we have Koblitz curves (which are non-supersingular) that are a class of computationally efficient elliptic curves. Several of these curves are recommended for cryptographic use by NIST. Supersingular curves are useful for cryptographic goal, too. In the 1,2,3,4 you can find several details.