Receive facing forward whenever you can. Check to create the angle, open the hips and shoulders to the target before the ball arrives, and direct the first touch into space away from the marker, then play forward.
A midfielder sits between a server and a target with a passive marker behind. The server passes in; the midfielder checks to the ball, opens up, receives on the half-turn away from the marker, and plays forward to the target. Reverse the direction so both sides are trained. Insist on a shoulder check before the ball arrives.
Receiving square is the safe option and that is exactly the problem. A player who takes the ball with his back to the opposition goal cannot lose it forwards, cannot be caught turning into a tackle, and cannot be blamed for much. He also cannot play forward, which means the safest body shape in football is the one that most reliably kills an attack.
The half-turn is a decision made before the ball arrives, not a technique applied when it does. The hips and shoulders open towards the target while the pass is still travelling, so the player is already facing the direction he wants to play by the time he touches it. Turning after receiving costs a second, and a second is enough for the marker to arrive.
Which makes the check the enabling action. Receiving flat-footed leaves no angle to open into, so the player moves to create one, usually a short movement away and then back. That movement also buys the yard of separation the open touch requires.
The first touch then goes into space away from the marker rather than dead under the body. Directional control is the whole skill, and it is what separates a player who can receive from one who can merely stop a ball.
Name the trade-off honestly, because players feel it and coaches rarely say it. Opening up does lose the ball more often, particularly at first. A player facing forward with a defender behind him is genuinely more exposed than one shielding it. That is the price of playing forward, and a coach who punishes the turnover will get square passing back within a week.
Save it to your private drills, then change the animation, adjust positions, swap equipment, or drop it into a session block. Or start from a blank pitch and build your own.