Interpretation of Helmholtz energy and Gibbs energy

I know that this question has many other variations on this site, but I'm trying to see if I understood Helmholtz and Gibbs energy properly or not. The material I'm reading from is ***Physical chemistry*** by Thomas Engel and Philip Reid, third edition, and ***An introduction to Thermal Physics*** by Daniel Schroeder.


In Schroeder's book, Chapter $5$, the author defined Helmholtz energy, $A$, as $A= U-TS$, where,


$U$ is internal energy of the system and
$S$ is the system's final entropy.


The author says that, *It is the total energy needed to create the system minus the heat you can get free from the environment at temperature $T$*. He further states that it is the available or "free" energy.


Then, for a system in constant pressure $(P)$ and temperature $(T)$ environment, he defines Gibbs energy, $G$, as $G= U-TS+PV$, where $PV$ is the atmospheric work term that's in enthalpy,


$H$ $=$$U+PV$.


Also, from Engel and Reid, chapter $6$, we have, for isothermal process,


$dA$ $\le$ $đw\_{total}$ $...$$(1)$,
including expansion and non-expansion work, where the equality is satisfied for reversible process. Equation (1) allows us a way to calculate maximum work that a system can do on the surroundings.


And in a similar manner $dG$ $\le$ $đw\_{non-expansion}$ $....$$(2)$, where the equality is satisfied for reversible process.


It is stated that equation $(2)$ allows one to calculate maximum non-expansion work that can be produced.




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Now, this is my understanding:


For reversible process, $A$ represents total available internal energy, and a part of $A$, which is $G$, is the available energy to do non-expansion work.For irreversible processes the inequalities (1) and (2) gives the lower bound for expansion and non-expansion work respectively. Is it the correct way of understanding or am I way off?
Also, if $A$ does represent total available internal energy for reversible process then what does it represent for irreversible process? Thanks in advance


