We first met Ammon and Moav in Sefer Bereishis (19:36-38), had a rough encounter with Moav in Sefer Bamidbar (22:2-4, 25:1-3), were told in Sefer Devarim (2:9/19) not to wage war with either when passing near their land, and were commanded (23:4) not to accept them as full converts. Based on our passing near their lands before we crossed the Jordan River, we know that their land was on its eastern side. Based on which land Sichon conquered from them (which we then conquered from Sichon), we have a pretty good idea where Ammon and Moav were, with Moav (or what was left of it) south of Nachal Arnon (Wadi al-Mujib, which flows west into the middle of the Dead Sea) and Ammon (or what was left of it) east of the land conquered by Moshe and the Children of Israel, as well as north of some of it.
How far south Moav extended isn’t as clear. The commentators who lived in the Middle Ages assumed that Moav was south of the Promised Land, east of Edom (see Rashi on Bamidbar 34:3 and the map in Tosfos on Arachin 15a). Maps based on these commentators (e.g. Eileh Masay and Shaaray Aharon) show Moav east of Edom, extending north to Nachal Arnon. [Putting Moav that far south allows Wadi al-Hasa, which flows primarily west into the southern end of the Dead Sea, to be associated with Nachal Zered.] Scholars—including frum ones—have Wadi al-Hasa as the southern boundary of Moav, with Edom south of it (preventing Wadi al-Hasa from being Nachal Zered, as I discussed at the end of Sefer Bamidbar).
Ammon isn’t mentioned when the king of Moav sent for Bilam to curse the Nation of Israel, and the switch from plural (“לא קדמו”) to singular (“שכר”) in Devarim 23:5 indicates that only Moav was involved in hiring Bilam. However, Ammon isn’t mentioned regarding the land Sichon conquered either (Bamidbar 21:26-29), implying that Ammon was too strong for Sichon to conquer (21:24), yet the king of Ammon told Yiftach (Shoftim 11:13) that Israel had taken his land. Since he was referring to the land Moshe had conquered from Sichon, which was what Sichon had conquered from [Ammon and] Moav, it would seem that Ammon not being mentioned doesn’t necessarily mean that their land wasn’t conquered. By the same token, Ammon not being mentioned when Bilam was hired doesn’t necessarily mean they weren’t involved. But if they were involved, why wasn’t Ammon mentioned (in either situation)? [That Sichon took land from Ammon too—land that was then conquered by Moshe—is clear from Yehoshua 13:25 and from Gittin 38a.]
A similar question arises regarding Yiftach’s answer to the King of Ammon (Shoftim 11:17), as Moshe asking Moav for permission to pass through their land (and their refusal) is referenced, without mentioning Ammon. If Ammon didn’t refuse too, why would Moav’s refusal be relevant to Ammon’s claim? As a matter of fact, Moav is mentioned six times in Yiftach’s response (11:15-27), with Ammon only referenced at the beginning (when Yiftach denied taking the land of either Moav or Ammon) and at the end (when Ammon’s complaint was referenced). Why did Yiftach consistently refer to Moav if it was Ammon that was making the accusations?
Raavad, in his commentary on the Sifra (Metzora 5:2, on Vayikra 14:34), suggests that since Ammon and Moav were “brothers,” one king ruled over both, alternating which country the king came from. In Moshe’s time, when Sichon conquered territory from both countries, the king was from Moav, so only Moav was mentioned, whereas in Yiftach’s time, the king was from Ammon, who was representing both Ammon and Moav.
One of the weaknesses of this approach is that after Sichon conquered a strip of land between the two countries, their land was no longer contiguous, making it difficult for one king to rule over both. Another weakness is the hiring of Bilam being said in the singular—if the king was acting on behalf of both countries. Perhaps this is because no one from either country offered nourishment, whereas the act of hiring Bilam was a decision made only by those in charge, which was Moav.
In his commentary on Shoftim (11:13), Malbim seems to endorse Raavad’s approach. Nevertheless, in his commentary on Devarim (23:4) he says that Ammon and Moav were originally just one country, ruled by Moav (since they were the overwhelming majority), which eventually split into two separate nations. When Sichon conquered their land it was still one country, so only Moav (which is how the country was referred to) is mentioned, both in Bamidbar and with Yiftach. I would take it a step further, suggesting that Sichon conquering the strip of land in the middle of the combined country known as Moav caused it to be split into two separate countries. The descendants of Ben-Ami—the B’nei Ammon (see Bereishis 19:38)—lived in the northern section of the original “Moav,” while the descendants of his older brother, Moav, lived in the southern section. After Sichon conquered land in the center, the northern part became its own, separate country (Ammon), while what was left in the south was still called “Moav.” [It was therefore just Moav, i.e. the southern country, that hired Bilam—hence the singular verbiage.]
Aside from explaining why the Torah only mentions Sichon conquering land from Moav, as well as why Yiftach primarily referenced Moav (since until then it was only called “Moav”), this also explains why Ammon is referenced as a separate country shortly after Sichon conquered their land (e.g. Devarim 2:19). It would also mean that the area where the Children of Israel camped after conquering land from Sichon was called Arvos Moav because it had been part of the larger “Moav” (which included Ammon, Moav and the land conquered by, and then from, Sichon), not because it had originally been part of the country that was now only south of the Arnon. The king of Ammon was demanding the land that his people had lived in when it was still called “Moav,” land that—as Yiftach pointed out—was taken from them by Sichon, not by the Children of Israel.
Rabbi Dov Kramer wrote a weekly dvar Torah from 5764-5776, most of which are archived at RabbiDMK.wordpress.com and AishDas.org/ta. His geographical pieces, including the one on Nachal Zered, are available at dmkjewishgeography.wordpress.com.