Would a piece that moved like a king – but could be captured (and could be moved en prise) – be considered a major or a minor piece? I could be wrong, but it seems like the major determinant of whether a piece is a major piece or a minor piece is whether a king and one of that piece can force checkmate on a lone king. Would this be possible with a king and one of these hypothetical pieces? (Or is that not actually the determinant of what is a major piece and what is a minor piece?)
This piece is widely used in different fairy chess variants. It appears under several names, including Commoner, Guard and Man / Mann.

While the table in the Wikipedia article also linked to in @Allure's answer states it's worth about four points, the article itself limits that value to the endgame:


  In the endgame, where there is usually little danger of checkmate, the fighting value of the king is about four points (Lasker 1934:73). In the endgame, a king is more powerful than a minor piece but less powerful than a rook. Julian Hodgson also puts its value at four points (Aagaard 2004:12).


On the other hand, the Wikipedia article about the fairy chess piece says


  A man is approximately equal in strength and value to a knight, generally.


That would make it a minor piece. It might depend on the board size (it's less effective on large boards, and worth much more on small boards) or the other pieces available to the players.




  it seems like the major determinant of whether a piece is a major piece or a minor piece is whether a king and one of that piece can force checkmate on a lone king


That's close to the definition I've heard: a major piece is one which can cut off the enemy king from one side of the board. Rooks and queens can do this, but this non-royal king cannot.

On the other hand, there is a basic situation where this piece is even stronger than a rook. King + rook lose against king + queen, but king + commoner is a rather simple draw.