"Timeless and ever-shifting, this psychic expanse is known as the Realm of Chaos, the Warp, the Immaterium or Warp space. It is a dimension parallel to our own, a universe devoid of consistency and unbound by the laws of time and space, a random, unstructured panorama of pure energy and unfocused consciousness. It is Chaos in its truest sense, unfettered by the limits of physics and undirected by intelligent purpose. Warp space is Chaos, Chaos is Warp space; the two are indivisible."
The Chaos Gods and the Warp are one and the same. Chaos is the Warp and the Warp is Chaos.
As for the gods only being omnipresent in their realms, this is true, but the only reason for this is because of areas dedicated to other gods. For instance, Khorne is the entirety of his realm, but obviously not the entirety of Tzeentch's. "Omnipresent in his realm" is an acceptable change to make to the gods, as it is a bit more precise. As for the Emperor, he is in a constant state of holding off all four gods, and does not really have a realm, so his is fine.
Skarbrand chinked the armour of Khorne's manifestation. All Chaos Gods have this, and it is more a representation of themselves at the "center" of their respective realms. However, it is still only a part of them. For instance, Khorne's manifestation is equally as much Khorne as the oceans of blood or mountains of skulls in his realm are. They are only perceived differently.