Energy change of reservoir in reversible weight process

[Sebi123]

In what follows I refer to the ideas of "Thermodynamic: Foundations and Applications" by Gyftoploulos and Beretta. The abbreviated form of my question is: In a reversible weight process,
Ω₁^R-Ω₂^R = E₁ - E₂ (see eqn. 6.18, p. 99) is transferred out of the composite of a system A and a reservoir R to the weight, where Ω^R is the available energy of a system A with respect to reservoir R and E is the energy of system A.
When the above equation is true, why does the book occasionally (problem 6.4, example 6.4 on p. 94) calculates the energy change of the reservoir as E₁^R - E₂^R ≠ 0 for the case of reversible weight processes?
Because, according to the equation above, E₁^R = E₂^R must be true (see below). Also, (DE₁₂^R)_rev^A should always be 0, p. 108/9. However, this quantity is used extensively in the discussion of entropy as if it was non-zero.
My proof of E₁^R = E₂^R:
Given two states, A₁ and A₂, of a system A and a reservoir R. Consider two reversible weight processes, α und β, consisting of two sub-processes, where A₀R₀ is the state where A and R are in mutual stable equilibrium. The only difference is the final state of the reservoir (R₁ in process β instead of R₂ in process α).

α: A₁R₁ → A₀R₀ → A₂R₂
β: A₁R₁ → A₀R₀ → A₂R₁

The first sub-process in each process transfers energy of the amount of available energy to the weight: Ω₁^R. The second sub-process in α and the second sub-process in β both transfer the available energy Ω₂^R to the weight, because the stable equilibrium state is unique (see proofs on adiabatic availability on p. 76 and regard the adiabatic availability of A+R as the available energy von A) and Ω₂^R is independent of the initial (or end state in this case) of the reservoir (statement 3 on p. 89). The energy balances for both processes are:

α: Ω₁^R-Ω₂^R = E₁ - E₂ + E₁^R - E₂^R
β: Ω₁^R-Ω₂^R = E₁ - E₂

Comparing both results, one gets that all states of the reservoir have the same energy: E₁^R = E₂^R
So: Ω₁^R-Ω₂^R = E₁ - E₂
The last equation corresponds to equation 6.18, p. 99.

 

[Sebi123]

Maybe I should rephrase the problem above. The short version is:

In a reversible weight process, in which system A goes from state A₁ to A₂ and reservoir R from R₁ to R₂, the energy transferred to the weight is
(W₁₂^AR→)_rev = E₁ - E₂ + E₁^R - E₂^R = Ω₁^R - Ω₂^R
This is equation 6.5 in section 6.6.4.

On the other hand, in section 6.6 the authors say:
"[A] weight process for system A from A₁ to A₂ is [...] reversible only if"
Ω₁^R - Ω₂^R = E₁ - E₂
(eqn. 6.7) This would mean that the reservoir does not exchange energy, as E₁^R - E₂^R. However, in example 6.4 they get E₁^R - E₂^R = -50 kJ ≠ 0.

Why do both equations contradict?