What kind of Entropy is considered here for a single bosonic Oscillator?

In thermofield dynamics, which constitutes an algebraic formulation of thermal quantum field theory, an entropy operator $hat{S}_beta$ for a single bosonic oscillator is introduced as

$$hat{S}_beta=-k_B,(hat{a}^dagger hat{a},ln,sinh^2theta_beta-(hat{a}^dagger hat{a} +1),lncosh^2theta_beta)$$

with the Bose-Einstein distribution $sinh^2theta_beta=langle hat{n}rangle_beta$ ($hat{n}=hat{a}^dagger hat{a}$) and the single-oscillator partition function $cosh^2theta_beta=Z_beta$. The entropy is obtained as expectation value with respect to the thermal vacuum state $vert 0_betarangle$ as

$$S_beta
=
langle 0_beta vert hat{S}_betavert 0_betarangle
=
-k_B,(langle hat{n}rangle_beta ln langle hat{n}rangle_beta -(langle hat{n}rangle_beta+1)ln Z_beta).
$$

To which type of entropy does the latter expression acutally refer to? I am used to von Neumann entropy as employed in open quantum systems theory, but when I derive von Neumann entropy for a single bosonic (harmonic oscillator) in a canonical thermal equilibrium state, I obtain

$$S_beta=k_B,beta,langle hat{H}rangle_beta+k_B,ln Z_beta$$

with $hat{H}=omega(hat{n}+1/2)$ for a single oscillator.

To make things even more puzzling for me, in Celeghini et al., Ann. Phys. 215, 156, (1992), it was shown, that

$$S_beta
=
langle 0_beta vert hat{S}_betavert 0_betarangle
=
-k_B sum_{n=0}^{infty} W_nln W_n
$$

with $W_n=dfrac{sinh^{2n}theta_beta}{cosh^{2(n+1)}theta_beta}$, which seems to be formally identical to von Neumann entropy. The coefficients $W_n$ are directly obtained by using

$$vert 0_betarangle = exp(-frac{1}{2}hat{S}_beta)sum_{n}vert n,tilde{n}rangle$$

to evaluate the expectation value $langle 0_beta vert hat{S}_betavert 0_betarangle
$. However, the coefficients $W_n$ seem not to refer to the standard Boltzmann weights of a harmonic oscillators, or at least I was not able to show their equivalence.

EDIT

The normalized thermal vacuum state $vert 0_betarangle$ in TFD is generally defined as

$$vert 0_beta rangle = dfrac{e^{-betahat{H}/2}}{sqrt{Z_beta}},sum_n,vert n,tilde{n}rangle$$

where I consider $hat{H}$ as given above for a single bosonic oscillator.