Let $X$ be a smooth projective irreducible curve.
My understanding is that a Kähler differential on $X$ is an $\omega \in \Omega_{k(X)/k}$ where $k(X)$ is a function field of $X$. The object $\Omega_{k(X)/k}$ has the structure of a $k$-vector space. Formally, it is defined as some quotient and informally it consists of elements $df$ for $f \in k(X)$ satisfying relations one would expect when they think about differentiation.
I am solving an exercise that involves a regular differential, which I believe is not the same.
A regular differential is a similar object except it consists of elements $df$ for $f \in A(X)$ where $A(X)$ denotes the coordinate ring of $X$.
Are these definitions correct?
If they are, then every regular differential is Kähler. If not, what are the correct definitions?
There's a few different things going on here. The original definition of Kahler differentials is for a map of rings $R\to S$: we get an $S$-module $\Omega_{S/R}$. With some relatively straightforwards manipulations, we get that for a multiplicatively closed subset $W\subset S$, we get that $W^{-1}\Omega_{S/R}=\Omega_{W^{-1}S/R}$, so modules of Kahler differentials are compatible with localization and we can sheafify this construction to get a sheaf of Kahler differentials $\Omega_{X/Y}$ on $X$ for any morphism of schemes $X\to Y$. Sections of this sheaf are typically referred to as regular differentials (because they're regular sections of the sheaf of differentials).
One may verify that if $X\to Y$ is a morphism of affine schemes where $X=\operatorname{Spec} A$ and $Y=\operatorname{Spec} R$, then the global sections of $\Omega_{X/Y}$ are exactly $\Omega_{A/R}$ corresponding to the Kahler differentials of the induced ring map $R\to A$.
A meromorphic differential form is a meromorphic section of $\Omega_{X/Y}$, and in your case, this is the same as an element of $\Omega_{k(X)/k}$. (This is true more generally for a morphism of integral schemes $X\to Y$: a meromorphic differential is an element of $\Omega_{k(X)/k(Y)}$.) In general, there are more meromorphic differentials than regular differentials: any regular differential is a meromorphic differential, and there are always (globally defined) meromorphic differentials on a curve, but there might not always be (globally defined) regular differentials: $\Omega_{\Bbb P^1_k/k}$ has no regular global sections, but plenty of meromorphic global sections.
In your definition of a regular differential form, it looks like you're mixing up the affine case and the projective case when you mention a regular differential as $df$ for $f\in A(X)$. Your definition would work for the affine case, but it fails in the projective case, no matter whether you pick $A(X)$ to be the ring of global sections of $\mathcal{O}_X$ or the "projective coordinate ring". Both of these definitions are wrong: for a projective irreducible variety over a field $k$ of characteristic zero, the first definition gives no regular differential forms (which is bunk - a non-hyperelliptic curve of general type over $\Bbb C$ certainly has regular differential forms, for instance) and the second doesn't even work right for $\Bbb P^1_k$.