Do all forms of energy have the ability to do work? According to this definition of energy: it is the ability to do work. That is $F×s$.
But except for K.E. and M.E, no other energy form can do work. So why is this definition in Physics?
Lets take an object to space, it has P.E. - but without any external force, it can't be converted into other forms. But according to the definition of energy, it has its own force.
 A: I think it was James Watt who famously noticed that if you boiled water in a kettle, it would lift the lid, and that this meant that heat could do work.
A: The First Law of Thermodynamics can be written:
$dE = \delta Q - \delta W$,
i.e. the change in energy is equal to the heat gained minus the work done. Work done is a function of force (and distance) only, so if your energy type cannot provide a force (e.g. chemical energy) then all transfer must be in the form of heat.
In your example, for an object to have potential energy, it must exist in some potential field in the first place: it is that field that does work on the object. There can be no potential energy without the same field also imparting a force, the latter is just the vector gradient of the former.
A: We associate energy with the ability to do work. Having heated water, say 1 litre with 90°C (with the specific for water heat capacity) we have some energy to heat a room.
How much do we have energy? This depends from the room temperature, than lower, than more energy we have. So energy is the difference between some state(s) of a subsystem to another subsystem. Take we hot water from the first floor to the second and in relation to a room in the first floor you have two states with higher energIe in relation to the subsystem room. If a fire burn up the room (it's an example only, sorry), the energy flow takes place from the room to the water. This happens because it seems to be a natural law, that all somehow connected subsystems come to thermodynamic equilibration.
So if we label something with energy we know that the could use it. And on the other side, than we declare that something contains energy we try to extract it. Best example is the nuclear fusion and fission.
After the edit of the question:
Kinetic and potential energy are not we only energies which able to do work. Heat, torsion, pressure, nuclear energy we claim to be energies. Work, done by wind, water, animals and humans (called physical work) is the expression of energy contain. Light is energy (solar cells) and so on.
If you dig deeper you cam to the point, that all energies (perhaps without nuclear energy) are bases on electromagnetic processes. The exchange of photons is the real process. Someone does not agree? Take any process and grab deep enough.
A: Every form of energy can be converted to kinetic energy as work, or already are kinetic energy themselves.


*

*Kinetic energy of another body can be transferred to do work

*Elastic potential energy can be stored in elastic objects and the objects have a tendency to regain their original shape - elastic force. That's the $F$ in your equation.

*Gravitational potential energy is in fact negative, by the equation $U_g=G\frac{Mm}{r^2}$. The equation $mgh$ is merely an approximation that is practically correct within a distance interval (mostly used near the surface of Earth). This form of energy can be used to do work and make an object fall. In your example, even when taken to space there is a minute gravitational force that will slowly accelerate your object, hence doing work.

*Chemical energy is potential energy stored in molecules or atoms. Chemical reactions can release a lot of gasses moving at high speeds, this is the kinetic energy of gas molecules. When these kinetic energy promote random, Brownian motion, it becomes thermal energy. When the kinetic energy causes all the molecules to rush at a certain direction this creates a pressure wave and hence airflow, or even shockwave. Now that is macroscopic kinetic energy no doubt.

*Electric potential energy is, like gravitational potential energy, is the an object has when it is in a field, in this case an electric field instead of a gravitational field. Electric potential energy converts into the kinetic energy of charge carriers, or electric energy. That's what happens when objects with like charges repel each other, and electric current is the flow of charge carriers, so again kinetic energy.

*Nuclear energy is like the nuclear version of chemical energy, again a form of potential energy. Nuclear reactions give rise to new matter, $\gamma$-ray and a lot of kinetic energy (often in the form of thermal energy).

*Thermal energy is the kinetic and potential energy of molecules moving, rotating or vibrating in a random manner.

*Wave energy is the energy stored in and transferred by waves. In mechanical waves it fluctuates between kinetic and potential energy, and is the superset of sound energy. In case of electromagnetic waves, it is the superset of light energy.


All these "types" of energy can be directly converted to, or already are kinetic energy, so no problem with the definition of work. The thing is, everything boils down to the kinetic and potential energy of something: molecules for thermal, atoms for chemical, nucleons for nuclear, charge carriers for electric, etc. These distinguishments are just a way for us to better communicate and know what is at hand, and the very same amount of energy can be referred to with a different name even when discussing the same object, only a different aspect or property of it.
However I don't think the ability to do work is a good definition of energy. It is indeed true that something can only do work if it has energy, but this definition does not address what exactly energy is, and its primary notion. You said you read it from Wikipedia, so I suppose you also read in the same sentence that "it is difficult to give one single comprehensive definition of energy"[sic].
