imho geometrically the most interesting one is obtained as a quotient $SO(5)/SO(4)$ where the Poisson stucture on $SO(5)$ is not the so-called standard one, but one determined by an element in the maximal torus (sometimes they are called twisted). This is the Poisson analogue of what is mentioned in Quantum symmetry groups of noncommutative spheres - Varilly, Joseph C. Commun.Math.Phys. 221 (2001) 511-523 (where only the quantum counterpart is developed), ie. the Poisson version of the Connes--Landi noncommutative 4--sphere. As I mentioned in the comment above $SO(4)$ is not a Poisson-Lie subgroup of a $SO(5)$ but only a coisotropic subgroup.
The symplectic foliation is very interesting. In fact you have a level function which is a Casimir, so that leaves are contained in the 3-dimensional spheres $t= const.$ (0-dim leaves when $t=0,1$). Inside such spheres 2-dimensional leaves correspond to the usual description of the 3-sphere as two solid tori glued together. Of course the rank drops down to zero in two copies of $S^1$.
I sort of got the impression this is the only way you obtain a Poisson homogeneous structure on $S^4$ starting from a compact Poisson-Lie group.
There is another interesting Poisson structure on $S^4$ coming from Poisson-Lie groups, not homogeneous one but as a double coset. There the symplectic foliation consists only of two leaves: a 0-dimensional one and a $4$--dimensional symplectomorphic to the standard one on $C^2$ (so a sort of "Poisson compactification"). But since I contributed to this maybe it is interesting only for me...
About the notion of Poisson homogeneous spaces $G/H$ of a Poisson-Lie group $G$ one may consider:
1) $H$ is a Poisson-Lie subgroup of $G$ (def. of Chari-Pressley, indeed);
2) the projection $G:\to G/H$ is a Poisson map;
3) the action of $G$ on $G/H$ is both a homogeneous action and a Poisson action.
We have $1)\Rightarrow 2) \Rightarrow 3)$ but none of the arrows is reversible. $2)$ is equivalent to $H$ being a coisotropic subgroup of $G$ and is characterized by the fact that there exists a point in $G/H$ in which the Poisson bivector vanishes. Remark that in general the trivial $\pi=0$ Poisson structure on $G/H$ is not necessarily Poisson homogeneous, as you seem to assume, unless the Poisson-Lie structure on the whole $G$ is trivial.