Ammon: the people east of the Jordan whom the Bible almost always calls "the sons of Ammon", traced in Genesis to Lot and settled north of Moab around their capital Rabbah
Ammon is the name of a people, not of a country. Out of its 106 appearances the name stands on its own only twice. Everywhere else the Bible says "the sons of Ammon", a family phrase that most modern versions simply turn into "the Ammonites". That is not how the other neighbors work. Moab (מוֹאָב) and Edom (אֱדוֹם) are names that can mean a country all by themselves, which is why the text can speak of the plains of Moab or of someone living in Edom. This name cannot do that. When the writers want the territory they have to add a word, and they say the land, the border, the cities or the capital of the sons of Ammon.
So the surrounding sentence is doing less work here than it does with Moab. The name points at the same population whether they are crossing the Jordan to fight, hiring chariots, paying tribute, sheltering refugees, or being addressed by a prophet. Even their ruler is "the king of the sons of Ammon", a king over a people rather than the head of a country, and the god they worship, Milcom (מִלְכּוֹם), is introduced as the god of the sons of Ammon.
The name carries a family story. Genesis closes the account of Lot's escape from Sodom by tracing this nation, and Moab beside it, back to Lot's daughters, which makes these neighbors relatives rather than strangers, and Israel is told on that basis to leave their territory alone. The kinship did not keep the peace. Ammon is the enemy in the Jephthah story, in the war that made Saul king, and in the campaign during which David had Uriah killed, and the prophets charge it with grabbing land along Israel's eastern flank. Even so, the oracle in Jeremiah 49 ends with a promise to restore its fortunes.
When one person rather than the whole nation is in view, a different word is used (עַמּוֹנִי), which is how the Bible labels an individual Ammonite soldier, official or wife.