I have found two different definitions of a quasi-Hopf algebra $H$: The first one assumes that there is an anti-endomorphism of H (called antipode) with certain properties and the other one assumes the antipode to be bijective.
Is there a reason for those different definitions? What is the more usual one?
This is not really an answer but rather some thoughts on a quite similar problem, related to the definition of quasitriangular hopf algebras:
There are actually two variants of the definition of "quasitriangularity" (and more generally of the notion of "almost-cocommutativity") met in the relevant literature: Some authors (see for example Kassel's book, ch. VIII, XIII or Montgomery's book, ch.10) include in the definition of quasitriangularity the requirement for bijectivity of the antipode, whereas other authors (see for example Majid's book, p.40, prop.2.1.8) prove that it is not necessary and bijectivity can be proved to be a consequence of the rest of the defining axioms.
So, i was wondering whether there might be a similar situation here. Since I am not a specialist in quasi-Hopf algebras i cannot say more. In any case it would be helpfull if you provided direct citations to the couple of definitions you are talking about.