The one with the diode on the switches the extend through the playfield is 100% wrong. It looks like it toasted the 100 ohm resistor too.

Each sling and pop bumper solenoid activation switch is supposed to have a 22uf 10V capacitor and 100 ohm resistor wired across them. Often, these have been cut off by operators when the components failed rather than replaced. The purpose of the capacitor is to give the solenoid a more solid, thorough fire to improve ball action. If you replace the caps, make sure you get the polarity of the capacitor correct as it does matter.

Slings have 2 switches extending up through the PF wired in parallel. There will only be one cap/resistor pair on each set of sling switches.

There are scoring/sound effect switches on the slings and pops as well. You can see one in the background on the left side of your second picture. These are in the switch matrix (like target and rollover switches) and are supposed to have only a diode wired into them.

You can see the resistors and capacitors illustrated on page 25 of the Firepower schematics on IPDB

You are correct, the resistor and capacitor are in series, wired in parallel with the switch. If you look closely at the switch, you will see the terminal where the resistor and capacitor are soldered together is just that, a terminal that doesn't connect to any other parts of the switch.

The slings and pops will work without the resistor and capacitor in place. They just improve the performance of the slings and pops.