The oldest surviving inscribed Roman treaty is the treaty with the Aetolians made in 212/211 (IG IX^2 2.241). Consider the following passage (ll.4-7):

εἰ δέ τινές κα τού̣-/ τ̣ων τῶν ἐθνῶν οἱ ῾Ρωμαῖοι πόλεις κατὰ κρα-/ τος λάβωντι, ταύτας τὰς πόλεις καὶ τὰς/ [χ]ώρας ἕνεκεν τοῦ δάμου τῶν ῾Ρωμαίων/ τῶι δάμωι τῶι τῶν Αἰτωλῶν ἔχειν ἐξέστω

If the Romans capture any cities of these peoples by force, let these cities and their territory be held by the Aetolian people as far as the Roman people are concerned.

The inscription, like many Greek documents from the Roman period, is clearly a translation of an original Latin document. The use of δᾶμος τῶν ῾Ρωμαίων and δᾶμος τῶν Αἰτωλῶν is a dead giveaway. The terms are obviously translations of the Roman legal term populus Romanus, the "Roman people," and by extension the "Aetolian people." The phrase is peculiarly Latin, and not used in Greek treaties: cf. the roughly contemporaneous treaty between the Athenians and the Spartans in 269/8 (IG II^3 1.912), which says simply Ἀθηναῖοι καὶ Λακεδαιμόνιοι, "the Athenians and Spartans." A treaty between the Romans and the Methymnaeans from 154 (IG XII 2.510) does the same thing, as does the treaty between the Romans and the Astypalaeans from 105 (IG XII 3.173).

This latter document, however, reveals some complications. The inscription (now lost) preserves the treaty, but first it prints the text of a decree of the senate enacting the treaty between the two states. This isn't unusual. Roman official documents in Greece often have these "framing" texts, usually either decrees of the senate or letters from a consul, and the Greeks had a habit of including every possible piece of Roman correspondence related to an event, no matter how irrelevant, on the same inscription, in keeping with earlier Hellenistic practice. The decree of the senate, conveniently, refers to the Romans and the Astypalaeans in the same terms as the treaty. However, not all Roman foreign documents do this. The two decrees of the senate for the Thisbaeans from 170 (SIG^3 646) refer to the Thisbaeans simply as Θισ-/[βε]ΐς, "Thisbaeans." The letter from Flamininus to the Chyretiae from the 190s (SIG^3 593) is addressed to the magistrates and the city of Chyretiae, and then goes on to refer to Flamininus' own feelings and those of the populus Romanus. The Lex de Insula Delo (CIL I^2 2500) from 58 refers to the populus Romanus as the operative power at Rome, but then simply insula Delus, "the island of Delos." The arbitration between the Genuates and the Veturii (CIL I^2 584)--not a treaty, but a similar document--simply refers to Genuates and Langenses.

Clearly a variety of terms were possible in official correspondence. Most commonly, Roman orators referred (at least in public, rather than senatorial, speeches) to the populus Romanus as if it were basically synonymous with the state. The magistrates and/or the senate could sometimes be added to the group, but ideologically populus Romanus was clearly the more important term, and the one that pretty much always appears when the Romans talked about the state with other Romans. Roman laws use populus Romanus as the term for the state, as do foreign treaties. However, consider the origin of these documents. Both laws and treaties (as well as declarations of war and peace, authorization for various religious rituals, exemptions from the law after 67, etc.) were passed by the people in the voting assemblies. The texts of laws and treaties were both written using the voice of the populus Romanus that enacted them. Laws were phrased as commands to the magistrates, whereas treaties were phrased as addresses from one populus to another. That is, these documents are not referring to the state as a nation-state as we might today, but like the way that Roman orators speak about the Roman state they're taking the voice of the operative civil group.

What, then, of documents originating from outside the assembly? Polybius in Book 6 notes that since foreign envoys usually go to the senate before they go to the people (if they ever do at all), foreigners often mistake the senate as the ruling group at Rome, since it's the only one they ever interact with directly. Decrees of the senate and documents like the arbitration of the boundary disputes between the Genuates and the Veturii take a number of different lexical standards. Some, like the decree on the Bacchanalia (CIL I^2 581) don't refer to any populus at all on either side, although it must be recalled that the inscription that we have is clearly excerpted and not the full text of the decree. Similarly, the decrees for the Thisbaeans don't mention the populus Romanus and don't refer to the Thisbaeans as a populus. It does, however, refer to the δημοσία πράγματα of the Romans, which is a very literal Greek translation of res publica. Again, however, consider the context of these documents. In both cases foreign embassies were coming to the senate and asking for their arbitration in their own civil affairs. The decree here is entirely from the senate, and unless something has to be enacted by the people (like a treaty) there's no reason to mention the populus. Moreover, the decrees for the Thisbaeans refer to the interests of the res publica as part of the (translated) formula "in keeping with the affairs of the Republic and his own good faith," the phrase that indicates the freedom of magistrates in the field or on their assignments to make calls as they believe them to be necessary.

In other words, the Romans didn't really have a word for the state. This is long before nation-states. Context here is king. The Romans sometimes refer broadly to res publica, but the term means "public affairs" and refers to anything from the actual function of the state to its interests. The Romans typically referred to the populus Romanus as the operative civil unit, and pretty much any act that was binding on the entire Roman state derived from the populus. But the populus wasn't necessarily directly involved in every activity of the state, and foreign communities could direct their correspondence to the senate or even to the individual magistrates or envoys dispatched to deal with their cases. In such cases the Roman representatives themselves tended to refer to the interests of the Republic or of the Roman people (see Flamininus' letter referring to his own opinion and the wishes of the populus Romanus), but this isn't like an official state seal or anything.

The problem becomes more complicated still when you consider, for example, boundary stones that might be inscribed "ex SC" ("by decree of the senate") or with the authority of the magistrate who set them up. But you get the idea.