First off, that's a really beautiful restoration. Who did it?

Second, it would be nice to see a shot of the inside of the door. Normally the silver rod is the coin reject rod and the red button starts the game.

Without seeing the inside, a guess would be that they have a coin switch hooked up to the reject rod and pushing that starts the game. It would not add credits, nor would it deduct them.

Does the game add credits for high scores/specials/match?

I can't see where the wiring that goes to the coin mech frame is leading. Does opening the slam switch on the door turn the game off?

So it adds credits. What happens when you hit the reject button with more than one credit on the game. Does it subtract credits?

Not going below 1 on the credit wheel is without a doubt just a switch adjustment on the credit unit. But why the reject rod starts a game but the start button doesn't is weird.

The original set up on "Cross Town" would have dual coin chutes.

Does the number of pins on the jones plug match the socket on the tilt board?

The small red button is supposed to start a game and subtract a credit. Normally, it will only operate if there are credits on the game, unless it's set to free play.
The long chrome button is the coin return, but it can be rigged to start the game. The left coin chute will start the game by tripping the coin switch. The right coin chute will add credits via the coin switch, but doesn't normally start the game. The number of credits it adds is set by the jumper on the motor board.

It's a little hard to tell from the pictures, but it looks like the way your coin switch is configured, it won't work on coins. The coin trip wire form should be configured so that it will push against the upper (longer) blade, and close the switch under the weight of a coin. The anti-cheat (heavier) wire form should be configured so that the lower bend is between the two switch blades as in this photo.

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Yours looks like it is configured so that pressing the coin return will either start the game or add credits, depending on how your Jones plug is plugged in.
I asked you how many pins are on your plug.
For a two-chute game, there are six pins on the plug.

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This is how the switch stack is supposed to be wired.
The black wire and the multi-striped one go to the normally closed slam switch. If this switch or circuit is open, the game will not power on.
The orange and brown wires go to the normally open start button.
It should start the game if the zero position switch on the credit unit is closed. Otherwise, it won't.

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Now normally the game should have dual coin chutes. The left chute should have a blue wire and orange wire on the coin switch. Closing this circuit should start a game, because it's like putting a dime in the coin slot. The right chute should have a green and orange wire on the coin switch. Closing this circuit will add credits, but won't start the game (usually multiple credits for 25c). The number of credits it adds is determined by the small plug and socket inside the game on the motor/relay board. Normally, because there are now credits on the reel, pushing the small start button should start your first game, and subtract a credit.
Again, depending on the plug your door has, how it's wired, and how it's mated to the female jones connector on the tilt board, will determine if the single chute your game has will either start a game, or add credits.
One word of caution; the slam switch wiring carries line voltage (110V), so be careful around that circuit.

Looking at your photos again, it looks like the switch stack is wired correctly. The way the coin chute wireforms are rigged, definitely looks like it should close the switch when you push in the coin return rod. That's one way to set a game to free play, but usually I only do this on add-a-ball games (which lack a start button). If you want the game to operate by coin, you'll have to rearrange the wireforms as in my photo, and of course you'll need a 10c coin mech.

Check the zero position switch on your credit reel. The brown and orange wires (start button) will only start the game if that switch is closed.

Yes, you say if you move the blue wire to the brown wire position, it will start a game. That's because completing the circuit between the orange wire and the blue wire simulates a coin entering the left chute. This will start a game regardless of whether there are credits on it or not.

Momentarily jumper the green wire to the orange wire. Does it add credits?

Ok. So now you have to figure out how the jones plug is wired. If it's stock, then you have to determine which wires go to the coin switch on your single chute; orange and blue (would just start the game), or orange and green (would add credits, and if the zero switch is closed, the small button should start a game and subtract a credit).

From your picture, it looks like orange and green, so your single chute should behave like the right hand chute in the original configuration. Closing the switch should add two credits, but not start the game.

The far left switch of the three in the picture (credit unit) needs to be closed for the small button to start a game. Yours may be dirty or open if the game doesn't start with the small button with credits showing on the reel.

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Yup. That'll do it!