Diu Fang

Diu Fang is a Chinese traditional game, apparently from the Central Asia region. In China it is known as square chess, game of squares, or playing squares (fang means square).

The game is played on a board with 7x7 or 7x8 intersections (but there are other sizes depending on the region),

The rules as stated by Olli Salmi:

• Each player has enough stones to fill half the intersections.
• The board starts empty and each player, in his turn, drops one friendly piece on an empty intersection. 
• When a player has four pieces making the corners of a square, he captures an enemy piece from the board. However, any piece forming a square cannot be captured.
• When all pieces are placed, pieces start moving to any adjacent empty intersection. Moves are mandatory, even if it means unmaking a square.
  
• Wins the player that captures all enemy pieces.

A Chinese text translation (from the previous link) states:

[...] The goal is to form a "fang" (square) by occupying one square of the board, which allows the player to capture one of the opponent's pieces. [...] Once all the pieces are placed, each player removes one of the opponent's pieces and begins moving their pieces. The objective remains to form squares and capture the opponent's pieces. Players can also dismantle and reform their existing squares to capture more pieces. The game ends when one player loses all their pieces, resulting in a loss.

"Diu Fang" has two variations: "Wei Fang" (aggressive strategy) and "Tian Fang" (peaceful strategy). In Tian Fang, players agree not to aggressively encircle the opponent's pieces, making the gameplay less confrontational and more harmonious compared to Wei Fang, which is more competitive and intense.

It is not stated if it is possible to form squares with 45º or other angles.

The game is reminiscent of Nine Men Morris and Alquerque (some boards have diagonal connections). It seems to have the same «win strategy», as in Morris, that allows the alternate movement of making and unmaking a square, thus capturing a piece in every two consecutive turns.

Details about this game (or family of games) are also described by Shimizu et al. in [1].

it is possible to note the diagonal lines at the board's middle

The authors state, besides the previous rules:

• The first stone must be placed at the middle (assuming a 7x7 board)
• "The players take turns in placing stones on the points of intersection of the lines, with the aim of creating a square around a field, while at the same time preventing their opponent from doing so."
  
• "The players take turns in taking a stone of the opponent which is not part of a square off the board and then moving one stone vertically or horizontally from a neighbouring point to the point left vacant by the removed stone. When a player manages to create a square of four pieces of his own he is allowed to remove an opponent’s piece from the board. This is repeated again and again. So a player aims to create squares himself, as squares of four stones are safe and no stone can be taken from it." This paragraph seems to mix two different rules. Each piece move means unconditionally removing an enemy piece. Then, an extra capture is possible after a new square if made. This would make for a much faster game.
  

The Ludii portal has some reconstructions that can be played (search for Fang).

In the main Fang variant, Ludii ruleset mainly follows [1]:

• 7x7 board, with diagonals in the square formed by the central 3x3 lines. 24 pieces per player. 
• Black plays first. Players alternate turns placing a piece on one of the empty spots on the board. 
• The first stone must be placed in the central spot. 
• They attempt to orthogonally surround an empty point with four of their pieces, making a square. 
• When all of the pieces are placed, players alternate turns removing one of their opponent's pieces from the board. They then move one of their pieces along the lines of the board to the place left vacant by the piece they removed. 
• When a player makes a new square, they may remove another of the opponent's pieces from the board. Pieces that are in a square around an empty point may not be captured, and captures are not made in the placement phase. 
• The player who captures all of the opponent's pieces wins.

This surrounding of an empty intersection does not appear at [1], which just says creating a square around a field. Anyway, this rule implies 45º angle squares. And this ruleset does not refer to any other type of square. So perhaps Salmi's description skips this detail, which seems probable to me (using different shapes of squares feels too modern).

[1] Shimizu et al., Game boards in the Longmen Caves and the game Fang, BGS 5.

[addendum] I found this Chinese site which says (using automatic translation):

Rules for the Game "Fangqi" (方棋)

Introduction. Fangqi is a collective term for traditional two-player board games popular in northern China. The capture method resembles that of Zhizi, where forming specific patterns allows the capture of opponents' pieces. Rules, board designs, and names vary slightly across regions.

Game Phases: 

1. Placing Phase: Players alternate placing one piece at a time on an empty intersection of the board until all pieces are placed.
2. Biqi Phase (Removal Phase): If the board is filled, each player selects one opponent's piece to remove. On odd-lined boards, the second player selects first, and on even-lined boards, the first player selects first.
3. Moving Phase: Players take turns moving one piece along a line to an adjacent empty spot.

Capturing Pieces:

• Captures occur when a specific pattern is formed (e.g., a square or rectangle). The capturing player removes a specified number of opponent's pieces, prioritizing unprotected pieces.
• Common patterns:
• Square: Four pieces forming a square remove one opponent's piece.
• Rectangle: Six pieces forming a rectangle remove two opponent's pieces.
• Victory is achieved by reducing the opponent's pieces to a number insufficient for forming capture patterns.

Movement Rules:

• Generally, players can only move one piece to an adjacent empty intersection.
• In some digital implementations, a player who cannot move loses; others allow returning to the Biqi Phase to remove additional pieces.

Regional Variations and Specific Rules:

1. Daxing District (Beijing)  
   - Placing Rules: Pieces cannot be placed at the four corners.
   - Capture Patterns:
     - Dragon: Five pieces in a straight line (horizontal or vertical) capture three opponent's pieces.
     - Square: Four pieces forming a square capture one opponent's piece.

 2. Guan County (Hebei)  
   - Placing Rules: Same as Daxing.  
   - Capture Patterns:
     - Dragon: Five pieces in a straight line add two pieces in the placing phase or capture two in the moving phase.
     - Square: Four pieces forming a square add one piece in the placing phase or capture one in the moving phase.

 3. Qinghai  
   - Unique Rules: Pieces are returned to the opponent instead of being removed.
   - Capture Patterns:
     - Bar (杠): Five pieces in a line remove three opponent pieces, with the line marked as unusable for further captures.
     - Square: Four pieces forming a square remove one opponent piece and mark the square unusable.

 4. Shandong Variations (Wugunqi, etc.)  
   - Capture occurs during the placing phase, but pieces are marked and removed before the moving phase begins.
   - Unique terms like "Five Tigers" and "Three Diagonals" describe specific formations.

 5. Northeast China (Manchu)  
   - Similar to Shandong, with unique formations such as "Big Tiger Claw" and "Sky Bar".

Summary. Fangqi incorporates a rich variety of rules that adapt to regional preferences. Core elements such as strategic placement, capture through pattern formation, and movement across intersecting lines remain consistent. Each regional variant adds unique strategies, fostering a diverse gaming experience.