I am aware of four states that used to style their chief executives as "president": Delaware, New Hampshire, Pennsylvania, and South Carolina. All of these positions were established in 1776, when the Thirteen Colonies rose in rebellion against the Crown and created new state constitutions to replace the old Colonial Charters. A majority continued the old title of governor, but some renamed their chief executives "president" in a cleaner break with monarchial tradition.
Governors were vice-regal appointees that oversaw colonies on behalf of the Crown. The title of president, meanwhile, had been used by English Commonwealth during the brief abolition of the English monarchy in the mid-17th century. The Founding Fathers would have been familiar with the title's history, and anti-royal charms. One can easily see the appeal in that revolutionary period.
There was initially no imaginable problem with this. Under the Articles of Confederation, the nascent United States had no executive branch: there was no "President of the United States" to confuse with the state presidents. Things changed when the Constitution was adopted in 1787, creating the office of presidency.
All of the state presidencies were renamed governors fairly soon after the creation of a federal presidency: Pennsylvania in 1790, and Delaware and New Hampshire in 1792. The one exception was South Carolina, which reverted back to governor much earlier in 1779. What they do have in common though, is that the switches were all implemented alongside changes to their respective state constitutions.
In a sense, therefore, state presidencies were a 1776 fad. The states updated the title as soon as was practically convenient.
Note that Samuel Adams was President of the Massachusetts Senate.