Although work and heat do the same thing (increase or decrease the internal energy of the system), There is still a fundamental difference between them. The definition of entropy necessitates the consideration of heat and not work. Second law of thermodynamics essentially states that heat cannot be converted to work completely by a cyclic engine. Is the word "Thermodynamic work" or "Hidden work" suitable for heat?
I describe a view here that goes back 200 years now to Carnot's "Reflections on the Motive Power of Fire," published in 1824, which after Colding and Mayer's discovery of energy conservation has largely been suppressed, misunderstood or ignored. Ove the intervening years many historians and physicists have claimed that Carnot was the originator of the concept of entropy, although his use of the words caloric (or calorique) and chaleur itself was sometimes ambiguous or contradictory.
Putting aside historical priorities, we can define thermodynamic work as the movement of thermodynamically relevant stuff by thermodynamically relevant force. Here the various stuff and forces are the subject of mechanics, electricity, magnetism, chemistry, etc. and the goal is to investigate how the movements of the various form of stuff and forces are related to each other.
In the simplest model of a thermodynamic interaction, we imagine that the system that can participate in several forms of interactions (gravitational, mechanical, chemical, electrical, etc.) are surrounded by pairs of work reservoirs of the same kind, a pair of mechanical, a pair of electrical, etc. Each reservoir is imagined to be so large and the stuff in the system being moved to and from to be so small as compared to the reservoir's size that the reservoir's characteristic potential does not change by the interaction with the system. We might call the reservoirs of infinite extent.
If in the $k^{th}$ interaction, an amount of stuff $\Delta X_k$ is moved from a reservoir storing an infinite amount of $X_k$ at potential $Y^1_k$ to another reservoir, also storing an infinite amount of $X_k$ now at potential $Y^0_k$ then, by definition, the amount of work $(Y^1_k$ -$Y^0_k)\Delta X_k$ was lost between them. This in complete analogy the way gravitational work is done on a mass of "human" size in the gravitational field of the Earth because the field is not affected by the work. Similar work can be done on electric charges in an electrostatic potential field or chemical compound in a chemical potential field, etc.
Now Carnot's calorique or what we nowadays call entropy is similar in the sense of being the stuff of work in that if an amount of entropy $\Delta S$ is moved from a "heat reservoir", better called an "entropy reservoir" at temperature $T^1$ to another at temperature $T^0$ then thermal work in the amount of $(T^1-T^0)\Delta S$ is lost.
The purpose of the thermodynamic system sandwiched between these pairs of reservoirs is to facilitate possible work interactions among these reservoirs by allowing that the system participate in all the possible interactions but not necessarily simultaneously. The usual Carnot cycle is an engine that can connect to a pair entropy reservoirs and a pair of other reservoirs, usually mechanical but can also be magnetic as used in paramagnetic cooling.
A single equation/inequality based on pure observation, physics and chemistry, summarizes almost everything with $k=1,2,..,n$ denoting the various nonthermal interactions:
$$\sum_{k=1}^n (Y^1_k-Y^0_k)\Delta X_k + (T^1-T^0)\Delta S =\mathcal D \ge 0 \tag{1},$$
where $\mathcal D$ denoted the total lost, i.e., dissipated work, that is never negative.
A reversible process is defined by the equality $\mathcal D=0$, and irreversibility obtains when it is not reversible that is $\mathcal D >0$. For a reversible transport of stuff $\Delta X_k$ and $\Delta S$ we see the the various lost work terms are in balance, just as a see-saw in a gravitational field, or any mechanical system with friction and other types non-mechanical "losses". You can also see that if one ignores certain non-thermal processes, interactions, that are actually happening, then it may appear as dissipation of the work of the other work processes that are being taken into account. This is the origin of one major branch of irreversible thermodynamics, where these dissipating processes are called internal or concealed, and I think this may be what you are referring to as "hidden work" in your question.
This concept of "hidden work" can also be combined with another major assumption, you may call it axiom, according to which the dissipation, $\mathcal D$, the total lost work is actually a constitutive quantity characteristic of the system itself through which the transport of the various work stuff is taking place.
So far, we have only discussed work conservation if $\mathcal D=0$ or work non-conservation if $\mathcal D >0$, this is the first half of the 2nd law. When this is rounded up with the 1st law we will also give meaning to $\mathcal D$. Specifically we can rewrite Eq (1) as energy conservation. Define $\Delta U^i = \sum_{k=1}^n Y^i_k\Delta X_k + T^i\Delta S;i=1,2$, then:
$$\Delta U^1=\Delta U^0 + \mathcal D\tag{2}$$
If we assume that the system has returned to its original state at the end of the interactions, then $\mathcal D$ ends in the reservoirs of index $i=0$, and all experience shows that it ends in the entropy reservoir at temperature $T^0$, and never in anything else. This is the other "half" of the 2nd law. Specifically, that if the system performs a cycle then the entropy reservoir at $T^0$ will absorb $$\Delta S^0= \Delta S + \frac{\mathcal D}{T^0} \tag{3}.$$
This Eq (3) is sometimes phrased as that the system generates entropy in the amount of $$\sigma = \frac{\mathcal D}{T^0} \tag{4}.$$
In this view of thermal interactions, "heat" is only the dissipation $\mathcal D$ and is "evolved heat"; "heat" or "evolved heat" is not work and it is not energy, per se. Instead, thermal work is the transport of entropy from one temperature to another, and entropy is a para-conserved quantity, meaning that the entropy participating in any process cannot decrease, it can increase in an irreversible process. Electric charge is conserved, gravitational mass is conserved, inertial mass is conserved, volume is conserved, surface area is not conserved, etc., but entropy is para-conserved, and each of these quantities has an associated work process.
Thermal work, that is entropy transport between two different temperatures can convert to other types of work but cannot convert if the transport is isothermal. This does not mean that during isothermal entropy transport the system cannot work just that the source of such work is not thermal.
It is not "heat" that cannot be converted to work completely but that entropy is a para-conserved quantity and thermal work is the transport of that para-conserved entropy between temperatures.
Of course, evolved heat, ie., dissipated work, can be used for thermal work but at another, lower temperature; entropy is just entropy, if its temperature dropped you move it to a lower temperature and then you obtain thermal work.
Lastly, for thermodynamic systems that cannot be reasonably described as being sandwiched between pairs of work reservoirs; the answer is still basically Eq (1). We break up the system into small pieces so that we have $Y^1_k$ and $Y^0_k$ infinitesimally close and the flow $\mathbf J_k dt \approx \Delta X_k$ being so small that its transport does not disturb the potential field $\nabla Y_k \cdot \delta \mathbf r \approx Y^1_k-Y^0_k$, and then $\mathbf J_k\cdot \nabla Y_k$ is the working (ie., work rate) per unit time and per unit volume lost in the $k^{th}$ interaction ($X_0=S, Y_0 = T$).
