Gottliebs don't switch on with the left flipper button as do Williams and some Bally machines.
This is a triple chute game; common for "King of Diamonds". Check the plugs on the motor board which select credits per coin. The game is supposed to start when pressing the small button in the center of the door if there are credits on the replay unit, or if the zero position switch on the replay unit is jumpered closed (free play).
The middle coin chute is 10 cents per play, so it's correct that tripping the coin switch will start the game without adding a credit.
Depending on where the "credits per coin" plug for the third chute is plugged, that coin switch should add "x" credits, but not start the game until the replay button is pressed.
As far as turning the game off, if either the weighted switch on the coin door, or the normally closed switch on the motor board are opened, the hold relay will release, effectively turning the game off. The recommended method is to give the bottom of the cabinet a sharp slap to turn off the game.
Often, the hold relay coil is bad, so the contacts on the relay are sometimes jumpered closed. It sounds like your hold relay is working properly because you say the anti-cheat switch turns off the game when opened.