A subset $D$ of group $G$ is called a $(v,k,\lambda)$-difference set if each element of $G$ (except identity) can be expressed as difference of $D$ elements ($d_1-d_2$) in exactly $\lambda$ ways. $v$ and $k$ are the orders of $G$ and $D$ respectively.
For any automorphism $\phi$ of $G$ the $\phi(D)$ is also a difference set. This gives us a family of equivalent (not sure if the term is correct) difference sets.
Is there a canonical form - a single set that represents the family? Like a form that you transform a difference set to, to understand which family does it belong to?
For example, in this database each of the families is represented by a single difference set - example. This leads to a hope there could be a canonical form, but is a definition of such known?
My main concern is commutative groups, but I would also appreciate a general answer for all difference sets as well.
There is no real canonical representative for a family of equivalent difference sets, but there is sometimes almost one.
For example, you are probably aware that for $q\equiv_4 3$, the set of nonzero squares in $GF(q)$ forms a difference set. An example is $\{1,2,4\}\in GF(7)$, a $(7,3,1)$-difference set. But as you note above, the automorphism $\sigma=(0123456)$ acts on this set to give $\{2,3,5\}$, which is easily seen to be an equivalent (yes, that is the correct term, since you mentioned it) $(7,3,1)$-difference set.
So in some sense, the set of nonzero squares in $GF(q)$ is the canonical difference set in $GF(q)$, but this is never explicitly stated anywhere. You just notice in the literature. I guess that when across all equivalent difference sets, one has a nice set of rules that can be remembered to generate it, it sort of tacitly assumes a canonical nature, though, to be deliberately annoying, that statement itself is hardly canonical.
Plus, it leads to silliness. Consider the 'twin prime power' difference sets in $GF(q)\times GF(q+2)$ where both $q$ and $q+2$ are prime powers. Then, the 'canonical-in-the-sense-that-I've-mentioned' difference set is the set of all $(x,y)$ with the following satisfied: $x\in GF(q)$, $y\in GF(q+2)$ and also at least one of the following satisfied: $y=0$, $x$ and $y$ are both squares, or $x$ and $y$ are both non-squares.
So the blind insistence on a difference set being canonically identifiable by a nice set of rules leads to some real silliness, in my opinion.
This is all just what I've noticed while writing my thesis, and I hope it is helpful. Please don't hesitate to ask for further clarification.
For more, I recommend Moore and Pollatsek's excellent and highly readable overview, simply titled "Difference Sets". Beth and Jungnickel are invaluable references to have as well, and both are mentioned in Moore and Pollatsek. Each has chapters in Design Theory survey texts that are full of great information, too, though the references in Moore and Pollatsek would be enough to keep one busy for many years.