When looking into the definition of a Pin group, it turns out that there are - at least - three different ones in the literature, and they do not agree --- but thankfully all yield the same Spin groups.
In detail, let $q:V\to K$ be a nonsingular=regular quadratic form on a finite dimensional vector space $V$ over a field $K$ of characteristic not $2$, and $C = Cl(q)$ its Clifford algebra. With $\alpha$ the principal automorphism of $C$, determined by $\alpha|_V =-\mathsf{id}_V$, and $t$ the principal antiautomorphism of $C$, determined by $t|_V=\mathsf{id}_V$, the Clifford norm is defined as $N(x) =t(\alpha(x))x$, for $x\in C$, in most sources, but by $N'(x)=t(x)x$ in Scharlau's 1985 book (see below.)
With $C^*_{hom}$ the multiplicative group of $\mathbb Z/2\mathbb Z$--homogeneous invertible elements in $C$, the Clifford group $\Gamma(q)$ consists of those $u\in C^*_{hom}$ for which $\alpha(u)Vu^{-1}\subseteq V$. The Clifford group maps naturally onto the orthogonal group $\mathsf O(q)$, due to the Cartan--Dieudonné Theorem.
Restricting $N, N'$, or $N^2$ to $\Gamma(q)$, each map defines a group homomorphism to $K^*$, and "the" Pin group has been defined as the kernel of either in various sources (In what follows I made up the notation to keep the notions apart, except for the first one.):
$\mathsf{Pin}(q) = \mathsf{ker}(N|_{\Gamma(q)})$ is the definition given in
Scharlau, Winfried: Quadratic forms. Queen's Papers in Pure and Applied Mathematics, No. 22 Queen's University, Kingston, Ont., 1969 iii+162 pp. and in
Knus, Max-Albert: Quadratic and Hermitian forms over rings. Grundlehren der Mathematischen Wissenschaften, 294. Springer-Verlag, Berlin, 1991. xii+524 pp.
It is also the definition in the original paper by Atiyah-Bott-Shapiro on Clifford Modules. However, these authors only deal with negative definite real forms so that $\mathsf{Pin}(q)$ coincides with $\mathsf{PIN}(q)$ in the notation proposed below.
$\mathsf{Pin}'(q) = \mathsf{ker}(N'|_{\Gamma(q)})$ is the definition put forward in
- Scharlau, Winfried: Quadratic and Hermitian forms. Grundlehren der Mathematischen Wissenschaften, 270. Springer-Verlag, Berlin, 1985. x+421 pp.
$\mathsf{PIN}(q) = \mathsf{ker}(N^2|_{\Gamma(q)})=N^{-1}(\{\pm1\})$ is the definition one finds in sources more concerned with real Clifford algebras and their applications in Geometry or Physics, such as
Lawson, H. Blaine, Jr.; Michelsohn, Marie-Louise: Spin geometry. Princeton Mathematical Series, 38. Princeton University Press, Princeton, NJ, 1989. xii+427 pp.
Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile: Analysis, manifolds and physics. Part II. 92 applications. North-Holland Publishing Co., Amsterdam, 1989. xii+449 pp.
Lounesto, Pertti: Clifford algebras and spinors. London Mathematical Society Lecture Note Series, 239. Cambridge University Press, Cambridge, 1997. x+306 pp.
Note that $\mathsf{Pin}(q), \mathsf{Pin}'(q)$ are subgroups of $\mathsf{PIN}(q)$ of index $1$ or $2$.
To make things even more confusing, sometimes competing definitions are used side by side, such as on the wikipedia page on Clifford algebras or on an earlier page here on mathoverflow.
The advantage of the "small" Pin groups is that they map onto the kernel of the Spinor norm in $\mathsf O(q)$, but whether one uses $N$ or $N'$ changes the sign of the Spinor norm...
The advantage of the "large" PIN group is that it maps onto the orthogonal group whenever $K^*/(K^*)^2 =\{\pm1\}$.
So, what should be the authoritative definition, or should we live with small and large Pin groups, in which case an author should specify very clearly which one is meant?
