Ah the unspoken question everyone likes to dodge because it's very tricky...as far as the ancient Egyptians were concerned the most catch all term for their country was Kemet (km.t) meaning black land - a reference to the incredibly fertile black Nile sludge that got dumped on their land during the inundation (the Nile's annual flood). The word 'Egypt' itself is etymologically Greek, possibly a bastardisation of the Egyptian jA.t which was another word for the black soil.

However the much more difficult question is whether ancient Egyptians thought of themselves as a combined nationality? km.t is sometimes uses in titulary and other texts to refer to Egypt but much more commonly you get references to either Upper or Lower Egypt. The king himself had the title nb tA.wy - lord of the 2 lands - which pharaohs took when the country was unified (or when they thought it should be unified under them) and they wore the double crown of upper and lower Egypt. The double crown consists of the white crown of upper Egypt (the hD.t) and the red crown of lower Egypt (the dSr.t).

Historically when Egyptians looked back at times when the country was not unified they refer to it as a time full of chaos and disorder. Most of the lamentation texts of the Middle Kingdom refer to the 1st intermediate period as a time when the world was upside down, when the rich were thrown out into the street, it was ok to rob people and servants were in charge. However this is possibly a propagandist twist to exaggerate the importance of the pharaoh in bringing balance to a unified country. Evidence from the Theban kings in the 2nd intermediate period suggests that they didn't consider the country rife with terror - only lower Egypt which they didn't control.

I'm spite of how the country is split though I would suggest there was always a coherent 'egyptian-ness' that applied. In the report of Wenamun from the start of 3rd intermediate period it is clear that Herihor in upper Egypt and Smendes in lower Egypt are Egyptian people whereas Tjeker-Baal is not. This is where race comes into the argument which makes things more fun, when non-Egyptian races were in charge of Egypt, by and large they had the same style of representstion - the trappings of being an Egyptian - with a few of their own cultural tweaks. The Libyans for instance kept naming conventions and sometime they are depicted with the feather on their head indicating they were chief of the meshwesh - an old Libyan title. The Kushites are more famous for their adoption and adaptation of Egyptian stylings but they still kept separate naming conventions.

Thereafter though things get really messy and I'm out of my area of expertise, the conquests from Persia, Greece, Rome plus the appearance of Christianity and the rise of the Copts all warp both what Egypt is, Egyptian means and what it all looks like. As far as ancient Egypt is concerned Egyptian wasn't a nationality defined by land boundaries but more by how the people presented themselves.

PS: be wary when researching the term Kemet online, it frequently gets mistranslated to 'land of the blacks' which is very different and a very very wrong interpretation. It leads to a fair bit of misunderstanding.