In general the names came about organically. They generally described both the nature of the states and the European titles of their first rulers. Men who had been counts in Europe established themselves as counts in the Levant. Men who had been princes established themselves as princes in the Levant.

Looking at Edessa, immediately prior to the arrival of the crusaders it was ruled by a former Byzantine officer, Thoros. Thoros had been appointed as the governor of Edessa by the Seljuq emir of Damascus. So Thoros was very close to the equivalent of a Count. Almost immediately after being made governor, Thoros made a bid for independence. He and Tutush fought for several years. By 1098 Thoros didn't like his position and so appealed to the crusaders for help. He was a fellow Christian after all. Baldwin of Boulogne answered the call, and was adopted as Thoros' heir. He almost immediately turned on his adopted father and claimed Edessa as his own. Baldwin then took the title of Count... apparently for two reasons. First it was a title he already held as Count of Verdun. No one could claim he was overreaching. Second the titles of count and governor were close enough to be practically synonymous. Baldwin was assuming and Frankifying an existing title and claim.

Antioch happened similarly. It had been the center of a Byzantine thema and ruled by a δούξ/doux (roughly a Duke). Of course Muslims had held control for several decades since the doux ruled there. When Bohemond I gained control of Antioch, he gained control of one of the ancient seats of a patriarch, and a city that had a history of semi-independence and local self rule. Bohemond was already a semi-independent ruler as Prince of Taranto. So like Baldwin he applied his existing title to a new area that kind of resembled his old lands in prestige. He could have declared himself Duke instead, but that would have implied Byzantine suzerainty, and he clearly wanted nominal independence.

Likewise the first Count of Tripoli, Raymond was Count of Toulouse. He died before capturing the city and confirming his claimed title. But his son Bertrand accepted the title and the nominal overlordship of the King of Jerusalem.

Jerusalem was the one that bucked the trend. The first ruler Godfrey never appears to take the title king. Instead he simply went by his European titles of Duke and Prince. But foreign powers consistently referred to the new territory as a kingdom. So when Baldwin succeeded his brother he took the title of King.