hi! while it's possible you'll get an answer here, this isn't always the sportiest of subreddits, so you might try x-posting to /r/etymology (they study word origins), or see if there are any sport history buffs in /r/soccer, /r/rugbyunion or /r/football

They are called backs because they are back. The fullback is the furthest back, and the halfback is part way back. Association football (soccer), rugby football, and Australian football also use those terms.

The peculiar thing about American football is that the term fullback and halfback are used only for offensive backs, while they primarily denote defensive positions for the other forms of football.

In 1906 the rules changed drastically. In prior years the forward pass was not allowed, a first down took 5 yards, and there was now limit to haw many players had to be on the line. The result is that the "flying wedge" play became popular, where most of the players would start in the backfield and rush forward before the snap. This tended to kill at least a dozen collegiate players a year, never mind the concussions and broken bones. President Roosevelt got involved to bring about the rule changes to improve safety.

edit: The new rules required that at least seven players begin on the line, which means that there are at most four backs. Often one of the backs lines up near the end of the linemen in order to balance the run/pass options leaving just three in the backfield.