I thought internal energy is only the energy due to molecular motion and chemical bonds, but here it says it's also the gravitational potential energy of the system! Is that true?
It ultimately depends on the context. The energy is the energy; the way the different contributions to the energy are named depends on how these contributions affect the dynamics of the system. In a thermodynamical context, the denomination "internal energy" usually refers to the energy due to chemical bonds and due to molecular motion (i.e. the vibration and rotation of the molecules); in particular, the energy due to chemical bonds is indeed the potential energy due to the interactions amongst the atoms in the molecules. On the other hand, the potential energy due to external fields, that due to the interactions amongst the molecules, that due to the interactions amongst the electrons and the protons inside the atoms and that due to the interactions amongst the nucleons inside the nuclei is usually not included in the internal energy. The first cannot evidently be named "internal". The second is still not enough "internal" to be called so. The last two are not called "internal" as the processes that are studied by elementary thermodynamics usually do not involve neither chemical nor nuclear reactions: these energies don't change in thermodynamical reactions and as such they contribute only to the mass of the elementary degrees of freedom of the substance. Of course, elementary thermodynamics is able to describe chemical processes as well. In this context, one prefers to drop the denomination "internal" and use instead more meaningful concepts, such that of the "free energy" of the system.
Excluding an external gravitational field, look at the variation in the radius of our sun:
Temporal solar size variations affects gravitational moments and consequently the distribution of mass and changes in the rotation of its mass, from core to surface. Related gravitational parameters (See Eddington-Robertson gravitational parameters) strongly contribute to the relativistic precession of planets. They set the amount of curvature of space-time per unit rest-mass and the amount of non-linearity in the superposition law of gravitation. Solar energy is fuelled by nuclear reactions in the center of the Sun which at the present epoch of time is to be considered a constant power source. It is the solar surface processes that govern energy release however as irradiance is there bottlenecked and furthermore the irradiance is seen to vary on a time scale from minutes to that of the 11-year cycle.
The gravitational field of the Sun must thus necessarily acts as an energy reservoir, storing or releasing energy in step with the emissive limitations at the surface. The gravitational reservoir adjusts according to the action of rotational moments with the solar radius being a major consideration.