Let $A$ be a finitely generated commutative $k$-algebra, where $k$ is a field. I have seen two definitions of the algebra of differential operators on $A$, which I will write as $D(A)$, and my question is: when are they equivalent, and why? Even just providing a reference would be great.
One definition I've seen is:
Let $\mathrm{Der}(A)$ be the $k$-linear derivations on $A$, and let $A$ act on itself by left multiplication. Then $D(A)$ is the subalgebra of $\mathrm{End}_k(A)$ generated by $A$ and $\mathrm{Der}(A)$.
The other definition I've seen is saying:
An element $T\in\mathrm{End}_k(A)$ is a differential operator of order $\leq n$ on $A$ if $[\cdots[[T,a_0],a_1]\cdots,a_n]=0$ for all $a_0,\dots,a_n\in A$, where the bracket is computed in $\mathrm{End}_k(A)$. Then we define $D(A)$ as the set of differential operators of some finite order on $A$.
I believe that the first definition, which I'll call $D'(A)$ is contained in the second definition, which I'll call $D(A)$. But it's not clear to me when the opposite inclusion holds.
My second big question, which you can feel free happily ignore if you know the answer to the first, is: what is the 'right' definition of $D(A)$ when $A$ is a non-commutative algebra over $k$? How do the above two definitions agree in this situation?
The two notions are equivalent in the characteristic zero (smooth! as pointed out by Mariano in the comments) case. The reason they're equivalent basically boils down to the Leibniz rule: $x\partial_x-\partial_xx=1$ in the ring of differential operators. Here's a sketch of the proof for the case of the $n$-dimensional Weyl algebra, defined by $$W_2=k\langle x_1,\dots,x_n,y_1,\dots,y_n\rangle/([x_i,y_i]-1,[x_i,x_j],[y_i,y_j]),$$ which is the ring of differential operators on $A=k[x_1,\dots,x_n]$:
Let $T\in \mathrm{End}_k(A)$ such that the $m+1$-fold commutator with any $m+1$ elements of $A$ is zero, but the $m$-fold commutator is not. Pick the following basis of $W$ as a left $A$-algebra: $y_1^{i_1}y_2^{i_2}\cdots y_n^{i_n}$. Use the list of basis vectors such that $\sum i_j=n$ to determine the $A$-coefficients of each of these terms in $T$ by setting the first $i_1$ of $a_j$ to be $x_1$, and so forth. Subtract the resulting linear combination of differential operators from $T$ to obtain a differential operator of order $\leq n-1$, and repeat. Eventually you have $T$ written as an element of the subalgebra of $\mathrm{End}_k(A)$ generated by $A$ and $\mathrm{Der}(A)$.
The as-far-as-I-know correct answer to your second question mostly comes out of quantum groups. Call an $R-R$ bimodule $M$ differential if it has a filtration by submodules such that each subquotient is a quotient of a direct sum of the $R-R$ bimodule $R$. Define the maximal differential submodule of $\operatorname{Hom}_k(L,N)$ as the $k$-linear differential operators from $L$ to $N$ for $L,N$ both $R-R$ bimodules. See V. A. Lunts, A. L. Rosenberg, Differential operators on noncommutative rings, Selecta Math. (N.S.) 3 (1997), no. 3, 335–359. I'd welcome any comment from any readers who know more about this angle, though.
In the positive characteristic case, there were several attempts to define the correct sort of differential operators. The Grothendieck differential operators are those corresponding to the second definition you list. The crystalline differential operators are those corresponding to the first case you list- although generally one constructs them as a sheaf first. There are also divided power differential operators- picking the "correct" version can be an interesting part of setting up the problems you want to attack. There's some nice discussion of the theory for the integer case at https://mathoverflow.net/questions/56860/explicit-ring-of-differential-operators-for-polynomial-algebras-over-the-integers.