I haven't heard of that definition being used for a major piece, but it's a neat observation. Such a piece (call it X) should be able to force a checkmate in a K+X vs K situation.
Say White's pieces are on d3 and e3, while Black's is on d5:
1) White plays Xd4, and Black's king must retreat. Say ...Kd6 is played.
2) White plays Ke4. Now Xd5 is coming, ...
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 ...
That is an interesting distinction theory. Although this king-like piece could deliver checkmate with the help of the king, I've always considered the difference to be based on the value of the piece. A king-like piece would only be worth the same as the other minor pieces.
The king is worth about 4 pawns if its loss didn't lose the game, which means it's neither a minor or a major piece, but something in between.
(Major pieces like the rook is worth 5+ pawns; minor pieces are worth 3.)