There are several reasons, as you’ve noted in the question - mainly, Bohemond was already a prince, so that’s the title he continued to use. He also wanted to imply that he was independent from both Jerusalem and Constantinople, but the title "prince" wouldn't necessarily convey that, and neither would count or duke. Essentially, the hierarchical way that we think of royal titles now didn’t really exist in the 11th century, so while “king” was hypothetically the highest title, a “prince” could have just as much power, depending on where you were. The specific circumstances in Antioch were also a problem, because there was a dispute over what the crusaders were supposed to do with territory that they captured that had previously belonged to the Byzantine Empire.

First of all, the problem of medieval titles: these days we have king/queen, prince/princess, duke/duchess, count/countess, lord/lady…but that sort of hierarchy has evolved over the centuries, where monarchies have become more ceremonial and less the day-to-day government. They had those titles in the Middle Ages too, of course, and emperors as well, but there wasn’t always such a clearly specific hierarchy. “Emperor” for example literally just means “power-holder” in Latin. "King" in Latin (“rex”) means “ruler”, “princeps” simply means “first”, “dux” is a “leader”, and “comes” means (as Duolingo amusingly reminds me) “comrade”. Even the Byzantine emperor was called “Basileos” in Greek, but that was the same word that they used for ancient Greek kings. So these titles don’t have the same rigid hierarchical connotations that they have now.

A duke or a count could easily be more powerful than a king. The Duke of Normandy was often more powerful than the King of France. The ruler of Brittany was usually a “duke” but sometimes they were called “count” and sometimes they even referred to themselves as the “king of Brittany”. In Italy they preferred the titles prince and duke, but there was also a king of Sicily. Could someone simply declare themselves a king, in Brittany or Sicily or Jerusalem or anywhere else? Did they have the authority to do that by themselves? In the 11th century the answer was generally no, you can’t do that by yourself, but the church would argue that they could grant that title.

That's what happened with Jerusalem. When the crusaders conquered Jerusalem in 1099, Godfrey of Bouillon didn’t want to be called “king”. It’s often said that he used the title “defender of the Holy Sepulchre”, but he didn't really call himself that. He just used his title from back home in Europe, where he was Duke of Lower Lorraine. That didn't mean Jerusalem was a duchy and that Godfrey was “duke of Jerusalem”, but he was still Godfrey the duke. His brother Baldwin had no objections to being called king, but on the other hand he had no title back home. So could he just decide to call himself king all on his own? Not really…Jerusalem was worthy enough to be called a kingdom, and so its ruler should be a king, but Baldwin needed the Pope’s approval and had to be crowned by the church.

In Antioch the situation was a lot different. I answered a question recently about the oath that the crusaders swore to Emperor Alexios. Did they promise to hand over everything they conquered to Alexios, or just the parts of Anatolia that had been captured by the Turks? Apparently Alexios only wanted Anatolia back, at least as far as Antioch, which had been Byzantium’s stronghold on their eastern frontier. Since Bohemond was already well-known to Alexios (as the Normans of southern Italy frequently attacked the Empire), it’s possible that Alexios wanted to buy him off with a fancy title and land, and the understanding may have been that Bohemond would rule Antioch with the Byzantine title of “doux” (the same word as “dux” in Latin - a duke, although more in the sense of a governor than an independent ruler).

The crusaders besieged Antioch and captured it in 1097, but then they were trapped there by a Turkish siege until June 1098, when they broke the siege and defeated the Turks. Alexios was on his way to help them while they were stuck there, but some crusaders had escaped back to Constantinople and told the emperor to turn back. Bohemond and the other crusaders felt that Alexios broke the oath, while Alexios thought that Bohemond had broken the oath. For several decades afterwards, Byzantine policy towards Antioch was that it was a vassal/subject of the empire.

So what exactly was Bohemond’s role there? Was he a governor, a Byzantine doux, or something else? Why not call himself a king? Why not count? Between June 1098 when they defeated the Turks, and January 1099 when the crusaders continued south to Jerusalem, Bohemond wasn’t the only one who claimed Antioch - Raymond of Toulouse also claimed it. Eventually Raymond was convinced to abandon his claim and the crusaders continued on towards Jerusalem. But even then, Bohemond didn’t really call himself anything, much less prince:

“It is not possible to date accurately the first use of the title of prince by a ruler of Antioch and we cannot, in fact, be sure that Bohemond I did use the title while in the Near East. He did not use any title linked with Antioch in the only two extant charters that he issued in the Levant as ruler of the principality. In the first, dated 14 July 1098, before he had full authority over the city, he named himself ‘Bohemond, son of Robert Guiscard duke of Apulia.’ In the second, which contains no dating clause, he termed himself simply ‘Bohemond.’” (Asbridge, pg. 130)

He may have started using the title “prince of Antioch” after 1105 when he returned to Europe looking for support against Alexios, who was still trying to force Bohemond to submit. (Alexios, by the way, always called Bohemond “doux”, never prince, presumably with the understanding that Bohemond was governing a Byzantine province - and he did eventually force Bohemond to submit in 1108.) So why “prince”?

“From the eighth century onwards, on the Continent, dux et princeps denoted a sub-royal ruler of a region such as Brittany or Aquitaine. In southern Italy the Lombard dukes adopted the title when the Lombard kingdom in northern Italy fell. Its use was then inherited by the Normans when they took over southern Italy, and in Capua a principality was established that maintained its status until its absorption by Sicily in 1135. Therefore, at the time of the Latin capture of Antioch, the Normans of southern Italy understood the title princeps to indicate an independent ruler with full jurisdictional rights over a region or territory.” (Asbridge, pg. 132-133)

In other words, southern Italy already had a long tradition of using the title "princeps" for an independent ruler, and Bohemond himself had already been using the title “princeps” for his territories there (he was “prince of Taranto”), and that’s what everyone called him during the crusade.

“By using the title of prince, the rulers of Antioch made a statement about the extent of their authority. The title was probably understood in the Latin East, as it was in southern Italy, to indicate that a ruler had full jurisdictional rights over his principality. Thus, through the use of this title the rulers of Antioch were asserting their autonomy from other powers in Western Europe and the Levant. More importantly, this would mean that the early princes of Antioch claimed independence from the Byzantine empire and the kingdom of Jerusalem.” (Asbridge, pg. 133)

It’s really only in hindsight that Jerusalem was sort of artificially imposed as the head of all the crusader states. It didn’t really work that way at the time. Antioch was a principality but Tripoli and Edessa were counties. Raymond of Toulouse ended up ruling in Tripoli instead of Antioch, and Tripoli was a county because Raymond was a count back home in France - the same reason Antioch was a principality. It didn’t really have anything to do with a feudal hierarchy with a kingdom at the top.

The main source for this is:

- Thomas Asbridge, The Creation of the Principality of Antioch (Boydell, 2000)

Other useful sources:

- Jonathan Harris, Byzantium and the Crusades (Hambledon and London, 2003) - for the Byzantine viewpoint on the Antioch dispute

- Thomas Asbridge, The First Crusade: A New History (Oxford University Press, 2004) - Asbridge's general history of the whole crusade

- Andrew D. Buck, The Principality of Antioch and Its Frontiers in the Twelfth Century (Boydell, 2017) - a more specialized study by Buck, who was one of Asbridge's students