A definition of defect groups In Rouquier's 2006 ICM talk (see here), he gives the following definition of the defect group of a block $b$ (definition 2.1.2)

A subgroup $D$ of $G$ is a defect group of the block $b$ if it is minimal among subgroups such that the restriction functor $D^b(\mathcal OGb)\to D^b(\mathcal OD)$ is faithful. $D$ is then unique up to conjugacy and it's a $p$-subgroup of $G$

(where we are in modular characteristic $p$)
This is not the usual definition that one finds, and in fact the definition in the paper does not come with a reference. My question is :

Are there some references that define defect groups in that way, and work with this definition ? Or at least some references that prove that this definition is equivalent to the usual ones ?

Ideally I would prefer references that do the first thing, i.e. just define defect groups the same way Rouquier does, and prove their first properties from this definition, and then at some point prove the other characterizations - but the second one is fine as well.
 A: I doubt that there are references that do the first thing, and I don't know any that do the second. Rouquier's definition is a basically a cute and slightly idiosyncratic reformulation of the standard definition of a defect group as a minimal subgroup $D$ such that every $\mathcal{O}Gb$-module is relatively $D$-projective.
I know this is not quite what you asked for, but here is a sketch of why Rouquier's definition is equivalent to standard definitions.
First, if $D$ is a defect group (or contains a defect group) then the standard theory of defect groups gives that the natural epimorphism (given by multiplication)
$$\mathcal{O}Gb\otimes_{\mathcal{O}D}\mathcal{O}Gb\to\mathcal{O}Gb$$
of $\mathcal{O}Gb$-bimodules splits.
So for any objects $X$, $Y$ of the derived category, the restriction map
$$\operatorname{Hom}_{D^b(\mathcal{O}Gb)}(X,Y)\to
\operatorname{Hom}_{D^b(\mathcal{O}Gb)}(\mathcal{O}Gb\otimes_{\mathcal{O}D}X,Y)\cong
\operatorname{Hom}_{D^b(\mathcal{O}D)}(X,Y)$$
is a (split) monomorphism of $\mathcal{O}$-modules. So the restriction functor between derived categories is faithful.
Second, if $D$ does not contain a defect group, then there is some $\mathcal{O}Gb$-module $M$ such that the natural epimorphism $\mathcal{O}Gb\otimes_{\mathcal{O}D}M\to M$ does not split as a map of $\mathcal{O}Gb$-modules. However, it does split as a map of $\mathcal{O}D$-modules. Taking the kernel, we get a short exact sequence
$$0\to L\to \mathcal{O}Gb\otimes_{\mathcal{O}D}M\to M\to0$$
of $\mathcal{O}Gb$-modules that splits over $\mathcal{O}D$, but not over $\mathcal{O}Gb$.
In other words we have a nonzero element of the kernel of the restriction map
$$\operatorname{Hom}_{D^b(\mathcal{O}Gb)}(M,L[1])
=\operatorname{Ext}^1_{\mathcal{O}Gb}(M,L)
\to\operatorname{Ext}^1_{\mathcal{O}D}(M,L)
=\operatorname{Hom}_{D^b(\mathcal{O}D)}(M,L[1]).$$
So the restriction functor beween derived categories is not faithful.
