Largely, those provinces were structures built on already-existing divisions and traditions that - to the Romans - had been there ab antiquo. For example, the borders of the province of Thracia largely overlapped with the borders of the previous client kingdom of Thrace that had been subject to the Romans. Similarly, the borders of the Aegyptus province were more or less effectively the entirety of the Ptolemaic Kingdom but taken under Roman rule. In both cases, many local institutions of administration continued more or less uncontinued; in Thrace local communities continued to be headed by strategoi and the main administrative divisions remained strategiai. The Romans added a few extra divisions here and there, but largely it was allowed to remain as it had always been. The Romans, absent the cartographic thinking so common in modernity, would not have thought of these things in terms of hard and immovable borders, but considered these matters in light of tradition, prevalent landmarks, rivers and other natural borders and important cities. To take a non-Roman example, when Xenophon describes armies on the march in the satrapy of Lydia in the Anabasis, he describes it much like one would describe a metro map; he remarks on which cities are passed through and where they are in relation to certain landmarks.

Thence he marched two stages, ten parasangs, to the Psarus river, the width of which was three plethra. From there he marched one stage, five parasangs, to the Pyramus river, the width of which was a stadium. From there he marched two stages, fifteen parasangs, to Issus, the last city in Cilicia, a place situated on the sea, and large and prosperous.

In the same sense, the Romans would not strictly speaking organize their provinces into big bordered maps, they would instead assign cities and collections of villages to the category of a certain province. The provinces exist for administrative purposes and you cannot tax mountains and rivers. In the same way, Roman borders would not themselves have been hard borders delineated on maps, they would have been marked by rivers and natural borders, and when they weren't, they would not have been borders at all as we understand them. If the Romans crossed into the Parthian Empire, they would simply understand that at some point, you reached cities which no longer were friends and clients of the Roman Senate and instead paid tribute to the Arsacid Dynasty. For example, during the crossing of Crassus into Parthian Mesopotamia, it is said that:

The country was destitute of [Parthian] men [during Crassus’ advance into Mesopotamia in 53], but they [the Roman scouts] had come upon the tracks of many horses which had apparently wheeled about and fled from pursuit (οἷον‎ ἐκ‎ μεταβολῆς‎ ὀπίσω‎ διωκομενων‎). Wherefore Crassus himself was all the more confident, and his soldiers went so far as to despise the Parthians utterly, believing that they would not come to close quarters.

[...]

But such was the exceeding vanity of the man [Antony] that, in his desire for fresh titles of honor, he longed to have the Araxes and Euphrates [rivers] inscribed beneath his statues, and, without any pretext or design and without even a pretended declaration of war, just as if it were part of the art of generalship to attack by stealth, he left Syria and made a sudden attack upon the Parthians. The Parthians, who were crafty as well as confident in their arms, pretended to be panic-stricken and to fly across the plains (simulat trepidationem et in campos fugam). Antonius immediately followed them, thinking that he had already won the day, when suddenly a not very large force of the enemy unexpectedly burst forth, like a storm of rain, upon his troops in the evening when they were weary of marching, and overwhelmed two legions with showers of arrows from all sides.

Obviously, the sources here - Florus and Plutarch - are very biased and often somewhat questionable as sources, but they can serve excellently to illustrate the kind of thinking that went into this. Florus - the second source - especially remarks that "he [Antony] longed to have the Araxes and Euphrates rivers inscribed beneath his statues", remarking not on any kind of provincial administration but instead drawing the border and implanting the idea of territorial sovereignty into the reader's mind with the use of the Mesopotamian rivers. Exactly because maps were not as precise and constantly checked as maps are today - because it was not possible - conflicts over the placement of cities, settlements, water access and similar were commonplace. Both in classical Greece, and the Mediterranean in general. The Second Punic War is an extremely illustrative example of this.

The city of Sagantum that caused the conflict between Rome and Carthage in the first place was a small settlement built on the other side of the Ebro, which violated the Roman treaty with the Carthaginians to respect their respective borders as the Ebro. Hannibal would have lost little by simply letting it exist, it did nothing to stop Carthaginian expansion in Iberia. However, because states did not trust each other and most diplomacy took the form of compellance diplomacy, in which states would seek to compel each other, not convince, Hannibal could not ignore the city and had to lay it to siege. Similarly, the Romans could not simply let Hannibal take the city even though Rome largely did not want to go to war, because that would be an insult to Roman honour, so the states had to go to war.

In another case, in the context of classical Greece, the casus belli that Thebes relied on to denounce and attack the Phocian League at the onset of the Third Sacred War was that they refused to pay a fine imposed on them by the Amphictyonic League - founded to protect the Oracle of Delphi - because they had cultivated sacred land on the Kirrhaean plain. However, this fine was far beyond the ability of the Phocians to pay, even if they attempted to deny that they had cultivated the land, they would never have been able to in the first place. Again, the diplomatic norm was diplomacy to compel, where the use of force was always a possibility between states.

Similarly, a treaty had been made between the Median Empire and Alyattes' Lydia to respect the Halys River as the border between the two states. Here was an undeniable natural border, and yet the two still felt the need to solidify it with a dynastic marriage that made Alyattes and Astyages brothers in law to bring the two dynasts closer to one another. The fact that such a pact had been made in the first place is an indication that it had been disputed and contested between them; so when Croesus makes an attack upon the nascent Kingdom of Cyrus, it can be assumed that he saw this as an opportunity to solve the dispute in a way favourable to him rather than relying on the treaty with Astyages, since he had no such treaty with Cyrus.

In summary, as an answer to the question of how the Romans drew borders - and by extension how ancient states in general drew borders - both foreign and domestic, the answer is that states largely did not and could not draw borders as we often see online on maps. Instead, they drew borders through the human geography of city locations and village communities combined with traditions and the natural geography of rivers, mountains, valleys and highlands. Disputing border claims made between one another by states was common and often a source of conflict, especially in the context of the Mediterranean system of diplomatic norms that evolved; one based on compulsion to the advantage of one's own state over others. States did not trust each other, could not communicate swiftly or effectively and often abrogated or broke treaties made between them in respect to borders in order to advantage their own state and revise the system in its favour.

Sorry for the answer being so long, I kept feeling unsure if it was adequate, and I still don't feel entirely happy with it, but here it is!

Sources:

-Pierre Briant; From Cyrus to Alexander: A History of the Persian Empire (2002)

-Nikolaus Leo Overtoom; Reign of Arrows: The Rise of the Parthian Empire in the Hellenistic Middle East (2020)

-Arthur M. Eckstein; Mediterranean Anarchy, Interstate War and the Rise of Rome (2007)