Here the problem is that you are using too many notations and you are changing them constantly. 

1. The master equation in electrostatics : 
$$ \vec{D}= \epsilon_0 \vec{E} + \vec{P}.$$ 

i. Here $ \vec{D} $ is known as the **electric displacement vector**. This represents the electric field in a system due to the **free charges**. For example the a charged conductor or ions embedded in a dielectric material are the free charges of this system. We can control these free charges and consequently we have full control over $ \vec{D}$.

ii. $ \vec{P} $ is the **polarisation vector**. It is defined as the electric dipole moment per unit volume of the system. Due to the existence of free charges in the system, the **free or external or applied or electric displacement field**  $ \vec{D}$ polarises the atoms in the system by creating tiny dipoles. These tiny dipole moments constitute this polarisation vector. Also note that these tiny dipoles create the **bound charges** in the system. The value of these bound charges can be obtained from the polarisation vector itself. 

iii. $ \vec{E}$ is the **total electric field** of the system. Means it’s the field due to both free and bound charges present in the system.

iv. $\epsilon_0$ is obviously the permittivity of free space. Now for **linear dielectrics** (Dielectrics in which polarisation varies linearly with applied electric field,  $ \vec{D}$.), the defining equation is,
$$ \vec{P} = \epsilon_0 \chi_e  \vec{E}. $$

Note that in RHS we are putting total field $ \vec{E}$ not $ \vec{D}$. This is the definition. By using this convention everything works out well. This also leads to the equation $ \vec{D} = \epsilon \vec{E}  $. Where $ \epsilon $ is the permittivity of the dielectric material. 

Exactly similar quantities appear in magnetostatics too.

1. The master equation in magnetostatics: 
$$ \vec{H} = \vec{B}/\mu_0 - \vec{M}. $$

i. Here $ \vec{H} $ is known as the **external or applied or magnetising field**. This represents the magnetic field in a system due to the **free currents**. For example the a constant current carrying wire embedded in a paramagnetic material provides the free current to this system. We can control these free currents and consequently we have full control over $ \vec{H}$.

ii. $ \vec{M} $ is the **magnetisation vector**. It is defined as the magnetic dipole moment per unit volume of the system. Due to the existence of free currents in the system, the **free or external or applied or magnetising field**  $ \vec{H}$ magnetises the atoms in the system by creating tiny magnetic dipoles. These tiny dipole moments constitute this magnetisation vector. Also note that these tiny dipoles create the **bound currents** in the system. The value of these bound currents can be obtained from the magnetisation vector itself. 

iii. $ \vec{B}$ is the **total magnetic field** of the system. Means it’s the field due to both free and bound currents present in the system.

iv. $\mu_0$ is obviously the permeability of free space. Now for **linear magnetic materials** (materials in which magnetisation varies linearly with applied magnetic field,  $ \vec{H}$.), the defining equation is,
$$ \vec{M} = \chi_m \vec{H}. $$

This also leads to the equation $ \vec{H} = \vec{B}/\mu  $. Where $ \mu $ is the permeability of the material.  Now let’s come to your problem.

> A sphere of radius R of a linear magnetic material of permeability μ is
> located in a region of empty space where a uniform magnetic field B0
> exists. a) Knowing that the magnetization that appears on the sphere
> is uniform, calculate M, the dipole moment induced in the sphere and
> the B field at all points in space.

Here applied or magnetising field $ \vec{H} $ is given as $B_0$. Using the equations mentioned above you need to find magnetisation vector $ \vec{M} $ and total magnetic field $ \vec{B}$. As you have mentioned the magnetic field inside the material due to the constant magnetisation vector only is $ - \vec{M}/3$. Therefore in this case magnetising field inside the material is $ \vec{H} = \vec{B_0} - \vec{M}/3$.

So putting these in the master equation and using $ \vec{H} = \vec{B}/\mu $ we get,
$$\vec{B_0} - \vec{M}/3 = (\mu (\vec{B_0} - \vec{M}/3)/\mu_0 - \vec{M}. $$

Simplify the above equation to get your desired result.