The usual definition of a twisted Poisson algebra involving a 3form does not seem to refer to a Poisson algebra that is then twisted, i.e. twisting either the commutative product or the Lie bracket. Have I missed where such a discussion occurs?

1$\begingroup$ At @RicardoAndrade's request, I have added the tag sg.symplecticgeometry. I have also changed the tag "poissonalgebra" to "poissongeometry". $\endgroup$ – Theo JohnsonFreyd Nov 13 '13 at 1:23
I'm sure experts will be able to provide more detailed answers. My reading of the paper http://arxiv.org/pdf/math/0107133v2.pdf introducing twisted Poisson structures indicates that it is the Courant structure on the generalized tangent bundle which is twisted by the 3form, and that "twisted Poisson structures" are bivectors defining Dirac structures for the twisted Courant structure.

$\begingroup$ Yes, more precisely the bracket is twisted by adding a term $\endgroup$ – Jim Stasheff Nov 17 '13 at 21:23
The (algebraic) Poisson counterpart of Drinfel'd twist deformation from Hopf algebras is the following: let $u$ be an invertible element of a Poisson algebra $A$. Then $A$, with the same associative multiplication but with the new bracket defined by $$ [a, \, b]_u := u^{1} [ u a, \, u b] $$ is a Poisson algebra called the uconformal deformation of $A$. I deal with this deformation ( http://arxiv.org/abs/1406.3529 ) and seems to be quite intersting. The geometric meanning of it due by C.M. Marle (1991). A very tempting and difficult question that arise in the context is the classification of all uconformal deformation of a given Poisson algebra.