You probably know all that I'm going to say, but I'll mention it anyway, for any other interlocutors:
I'm going to adopt Stasheff's recommendation ("Drinfel$'$d's quasi-Hopf algebras and beyond", same volume) that "Drinfel$'$d-twisting" be called "skrooching" (a transliteration and abbreviation of the Russian, which is roughly four syllables.). Also, I'm going to talk about associative algebras and quasicoassociative coalgebras, not the other way around, even though the reconstruction theorem lands in "co" territory: if you really want to do Majid's theorem, you have to think about the convolution multiplication.
Anyway, then the basic Drinfel$'$d idea is that you start with something that I think should be called a "weak bialgebra": an associative unital algebra $A$ and an algebra homomorphism $\Delta : A \to A\otimes A$. This gives a functor $\otimes: A\text{-rep} \times A\text{-rep} \to A\text{-rep}$, which forgets to the usual $\otimes$ on vector spaces.
A quasicoassociative bialgebra is a weak bialgebra along with a distinguished invertible element $\psi \in A^{\otimes 3}$ so that $(\text{id} \otimes \Delta)\circ \Delta = \text{Ad}_\psi \circ (\Delta \otimes \text{id})\circ \Delta$, where $\text{Ad}_\psi$ is the inner automorphism of $A^{\otimes 3}$ by conjugating by $\psi$ — then acting by $\psi$ gives a natural $A\text{-rep}$ isomorphism $(X\otimes Y)\otimes Z \to X \otimes (Y\otimes Z)$ — and there's a pentagon for $\psi$ that matches Mac Lane's pentagon.
Given any invertible element $\varphi \in A^{\otimes 2}$, you can skrooch the weak bialgebra $(A,\Delta)$ by $\varphi$ to get a new weak bialgebra $(A,\Delta^\varphi)$, which is the same algebra and $\Delta^{\varphi} = \text{Ad}_\varphi\circ \Delta$. If $(A,\Delta,\psi)$ is quasicoassociative, then $(A,\Delta^\varphi,\psi^\varphi)$ is, with: $$ \psi^\varphi = \text{Ad}_\phi \psi \quad \text{where} \quad \phi = (\varphi\otimes 1) \cdot (\Delta \otimes \text{id})(\varphi) \in A^{\otimes 3}$$ and $\cdot$ is the multiplication in $A^{\otimes 3}$. You can translate this pretty straightforwardly into the representation theory.
The easier observation is quasitriangularity. A weak bialgebra $(A,\Delta)$ is quasitriangular if it comes equipped with a skrooch $\rho \in A^{\otimes 2}$ such that $\Delta^\rho = \Delta^{\rm op}$. A (quasi)coassociative bialgebra is quasitriangular if additionally $\rho$ satisfies two hexagons. If you skrooch a quasitriangular bialgebra $(A,\Delta,\rho)$ by $\varphi$, then you get a new quasitriangular bialgebra $(A,\Delta^\varphi,\rho^\varphi)$, where $\rho^\varphi = \varphi^{\rm op}\rho\varphi^{-1}$ (multiplication in $A^{\otimes 2}$; $\varphi^{\rm op}$ is the obvious element with the two $A$s in $A^{\otimes 2}$ flipped).