I quote the paper you are referring to:
"You will recall from E&M classes that the magnetic field inside a magnetic material can differ from the externally applied field Bext and from an auxiliary field H$H$ which is often introduced. In particular the relation between B$B$, H$H$, and M$M$ is B = μ0(H +M)$B = μ_0(H +M)$, where μ0$μ_0$ is a constant. In this course we shall only consider “weakly magnetic” materials where M ≪ H ≃ B/μ0$M ≪ H ≃ B/μ_0$, and so the difference between B$B$,Bext $B_{ext}$ and μ0H$μ_0H$ will be negligible. From now on, we will just label the field in Eqs. (1) and (2) by B$B$ and ignore any fine distinctions."
In other words when the susceptibility is small in the context of the reference it does not matter which one, but I note that according HEINE, BYV. "THE THERMODYNAMICS OF BODIES IN STATIC ELECTROMAGNETIC FIELDS" it is possible to transform the equations $$ F=\int_{all space} dv\int_0^{\textbf{B}} \textbf{H} \cdot d\textbf{B}$$ $$S=\int_{allspace}dv \int_{0,T=const}^{\textbf{B}} \left(\frac{\partial\textbf{M}}{\partial T}\right)_{\textbf{B}} \cdot d\textbf{B}$$ to $$S=\int_{V}dv \int_{0,T=const}^{\textbf{B}_{ext}} \left(\frac{\partial\textbf{M}}{\partial T}\right)_{\textbf{B}_{ext}} \cdot d\textbf{B}_{ext}$$and $$ F=-\frac{1}{2}\int_{V'} dv\int_0^{\textbf{B}_{ext}} \textbf{M}_{ext} \cdot d\textbf{B}_{ext}-\int_{V} dv\int_0^{\textbf{B}_{ext}} \textbf{M} \cdot d\textbf{B}_{ext}$$ and without permanent magnets we have the simpler $$ F=-\int_{V} dv\int_0^{\textbf{B}_{ext}} \textbf{M} \cdot d\textbf{B}_{ext}$$ where $V$ is the region of the body.
