I'm trying to find a good source for the definition of degenerate matter to differentiate it from Fermi gases. For my research a good section on history would be nice. This question is more specifically about stars composed of degenerate matter.
Originally, "degenerate star" was just a regular phrase describing a star made of degenerate matter, which referred to the dense matter made of nearly completely ionized atoms. The "coining" was a joint effort by Eddington, Fowler and Milne in 1920-s, they developed a theory of degenerate stars, see Leverington's History of Astronomy (p.186):
"Arthur Eddington had explained that in a star the radiation pressure and gas pressure balance the gravitational attraction. He suggested, however, that, as the radiation pressure is proportional to the fourth power of the temperature, a star could explode if the temperature became too high. In a parallel development, he had also explained in 1924 that white dwarfs, like Sirius B, are made of atoms that are almost completely ionised, which enabled the atoms to be packed very close together into what was called degenerate matter. In the following year, Nova Pictoris was seen to be caused by a star swelling up and exploding.
"Ralph Fowler of Cambridge developed a theory of degenerate matter in 1926, in which he confirmed that a white dwarf is largely composed of degenerate matter, with only its outer layer remaining gaseous. As the white dwarf cools near the end of its life, more and more of this gaseous layer becomes degenerate and, eventually, the whole star consists of degenerate matter at a temperature of absolute zero. The star is then a black dwarf radiating no energy. Two years later, Arthur Milne of Oxford examined the properties of stars, not just white dwarfs, assuming that there was some degenerate matter in the centre of most of them."