I've been trying to find a clean definition what people mean by when they talk about diffuse ionized gas in the interstellar medium, but I couldn't find anything so far. Apparently it's supposed to be trivial.
Is it simply ionized gas, mixed with non-ionized gas in the ISM through the process of diffusion?
The "diffuse ionized gas" (DIG) is another term for the phase of the interstellar medium (ISM) usually called the warm ionized medium (WIM). With a temperature of the order $10^4\,\mathrm{K}$, but extenting to lower and higher temperatures, it is hot enough to keep hydrogen ionized, and various metals exist as low-ionization species, such as S II, N II, and O II, and even (weak) O III (e.g. Hill et al. 2012; Zang et al. 2017; Weilbacher et al. 2018).
Rough pressure equilibrium with the other phases of the ISM results in characteristic densities of $0.1\,\mathrm{cm}^{-1}$, but typical ranges are an order of magnitude to both sides.
A review of the DIG/WNM can be found in Haffner et al. (2009).
I don't think it means anything special. Take, for example, the claim at Wikipedia
An interstellar cloud is generally an accumulation of gas, plasma, and dust in our and other galaxies. Put differently, an interstellar cloud is a denser-than-average region of the interstellar medium, (ISM), the matter and radiation that exists in the space between the star systems in a galaxy. Depending on the density, size, and temperature of a given cloud, its hydrogen can be neutral, making an H I region; ionized, or plasma making it an H II region; or molecular, which are referred to simply as molecular clouds, or sometimetimes dense clouds. Neutral and ionized clouds are sometimes also called diffuse clouds. An interstellar cloud is formed by the gas and dust particles from a red giant in its later life.
As well as another statement
In all phases, the interstellar medium is extremely tenuous by terrestrial standards. In cool, dense regions of the ISM, matter is primarily in molecular form, and reaches number densities of $10^6$ molecules per $cm^3$ (1 million molecules per cm3). In hot, diffuse regions of the ISM, matter is primarily ionized, and the density may be as low as $10^{−4}$ ions per $cm^3$. Compare this with a number density of roughly $10^{19}$ molecules per cm3 for air at sea level, and $10^{10}$ molecules per $cm^3$ (10 billion molecules per $cm^3$) for a laboratory high-vacuum chamber.
