In (Lounesto, Ablamowicz; 2012; pg307) Hahn defines a Clifford algebra as follows (almost verbatim):
Let $M=(M,q)$ with associated bilinear form $b$ be a quadratic module over $R$. A Clifford algebra of $M$ is a pair $C(M)=(C(M),\gamma)$ such that
(a) $C(M)$ is an $R$-algebra and
(b) $\gamma:M\rightarrow C(M)$ is an $R$-module map satisfying $\gamma(x)^2=q(x)1_C$ and $\gamma(x)\gamma(y)+\gamma(y)\gamma(x)=b(x,y)1_C$ for all $x$ and $y$ in $M$, and
(c) $(C(M),\gamma)$ is "minimal" (in the sense of a universal mapping property) with respect to (a) and (b).
Now I have also come across the term of a 'universal Clifford algebra' in e.g. (Porteous, 1995; pg130). Is the above definition defining a 'universal Clifford algebra' or a general (i.e. possibly non-universal) Clifford algebra. In either case how would we change it so it defines the other?