SueQ: You cannot jump a stone and sandwich the stone you jumped with the stone you jumped with -- as the position on "the other side" will be vacant. You can however jump a stone and sandwich another stone, which will be removed from the board.

Googling on 'Hasami Shogi', reveals that there are many variations on the game, and they are all called 'Hasami Shogi'.

First hit is BoardGameGeek. Its rule summary doesn't mention winning with five in a row, but otherwise, the rules and the boards are identical.

Sensei's Library describesa game identical to the one on BK, except for a few details:

• the stones are placed on the intersections instead of the squares (trivial detail, doesn't effect game play)
  
• black starts, not white (trivial as well)
  
• first player must move "into the four middle rows".
  
• capturing a corner stone by flanking it on both sides.
  

It also mentions only 8 stones are placed in the initial setup, but that seems a typo as the image showing the initial position shows 18 stones.

It does however mention variations, and the fact there's no consensus on what Hasami Shogi is:

• Start with 9 or 18 stones.
• Winning by capturing all but one, or by creating five in a row.
• Moving like a (chess) rook, or just one square.
• Possibility of sandwiching diagonally (as in pente).
• Sandwiching one stone, or can more be sandwiched as well.
• Jumping over own stones or not.

Chessvariants describes 'Hasami Shogi' as being played with 9 stones each, and stones moving as rooks (no jumping). You win by reducing the opponent to a single piece. As variants, it mentions winning by reducing your opponent to four pieces and taking an edge piece by flanking it from any two adjacent sides.
It also mentions Hasami Chess, played on an 8x8 board with 8 stones each.

Games of Soldiers lets stones move a chess rooks (no jumping), captures by sandwiching horizontally or vertically, with edge capturing by any two adjacent pieces as an optional rule. The goal is to reduce the opponent to a single piece (no five in a row rule). It also mentions Mak-yek (or Apit-sodok) with 16 stones each and an additional capturing rule. And a hexagonal variant, Take.

Games and Puzzles uses rooks moves and jumps, and as goal making five in a row, either straight or diagonally.

Looking at the sites above, and a few more, I think we can draw the following conclusions:

• The game is played on a 9x9 board.
  
• It's usually played with 18 stones each, sometimes 9.
  
• Pieces move as chess rooks.
  
• In some variants, a piece may also jump over an adjacent piece. This is sometimes restricted to jumping only over a friendly piece.
  
• Capturing is done by sandwiching horizontally or vertically. Sometimes diagonally sandwiching is allowed.
  
• Some variants allow "edge" captures by flanking an enemy stone on the edge from any two sides.
  
• There are two goals: getting five in line (usually straight, sometimes diagonal), or reducing your opponent to one or less stones. Some variants allow either goal, other variants allow only one of the goals.