Advanced/Retarded Eddington-Finkelstein coordinates & Black/White Holes

The different coordinate systems cover different parts of the full solution, which are relevant to different sorts of objects.

The full Schwarzschild vacuum solution has four 4D regions, and 3D boundaries (event horizons) between them. They're conventionally labeled with roman numerals:

Region II is the black hole interior, IV is the white hole interior, and I and III are exteriors. Only one of these exterior regions is present in black holes that form from collapsing matter, and it's arbitrarily taken to be I. A white hole that was simply a time reversal of a black hole would also have only one exterior region. An "eternal grey hole" is the only sort of object in which all four of these regions would actually be physically relevant at once.

The $r>r^*$ part of the Schwarzschild coordinates covers region I or III (normally taken to be I) and the $0<r<r^*$ part covers region II or IV (usually taken to be II, if this part of the coordinate map is used, which it typically isn't). The Schwarzschild coordinates don't cover the event horizons at all; $r=r^*$ is a coordinate singularity. The lowercase $t$ and $r$ coordinates shown in this image are Schwarzschild coordinates in units where $r^*=1$. ($T$ and $X$ are Kruskal-Szekeres coordinates.)

Ingoing coordinates (Eddington-Finkelstein and otherwise) cover regions I and II (or III and II) and the boundary between them. Outgoing coordinates cover regions I and IV (or III and IV) and the boundary between them.