the slingshots are the kicking arms under the plastics just above the flippers. If those are firing you probably have multiple issues.
the two holes at the top of the playfield williams calls "eject holes" on the schematic. The outhole is also generically an eject hole, but they do call it the outhole.
terminology is kinda vague ... eject holes = kickout holes = saucers depending on manufacturer, who is talking and what they'd been drinking
slingshot almost always just mean a vertical arm that kicks the ball sideways from behind a rubber ring.
tmi
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if you're interpreting the schematic, hang in there ... it'll start making sense as you plod thru it. For example, if your ball in play lights don't work, the typical approach is:
1] find those lamps on the schematic. They are around A-3
2] note the thing controlling the lamps is called the "ball count unit"
3] try manually stepping the ball count unit and verify the lights work. If not, make sure no other switches need to also be closed.
4] if ya conclude the unit/lamps are fine but the unit isn't stepping, then ya look for the "ball count step-up" coil. wms is nice and tell you in a chart that it's at F-8 on the schem. Then you try and figure out what the circuit means and you start poking around with a voltmeter or jumper wire to find issues.
in the ball count unit case, what the circuit says is:
1] the ball count unit will step up when the last enabled player is active (player unit keeps track of current player, coin unit how many players enabled)
2] the step up pulse is coming from score motor cam 1A switch. The switch stacks on score cams 1-6 go down/up in timed sequence, so 1A will happen quickly after the motor turns on.
3] the ball index relay switch needs to close
4] if you have an extra ball, a switch opens to disable stepping the unit
5] an outhole relay switch needs to be closed
the ball index relay is wtf #1, so you go find that (chart says F-17) and you'll see that you either have to tilt the game or score 10/100/1000 points. So conclusion is the ball count unit won't normally move until you score some points. This was the EM way of doing a ball save ... if you shot the ball and scored nothing, you got the ball back to try again.
the fun part is when is the outhole relay switch at schem C-8 actually closed (connecting yellow to R-O)? It can get kinda ambiguous figuring out in what machine state the schematic is drawn. While there's some rules of thumb that usually apply, ya get bitten enough times to either go look further in the schem, look in the game or just make some logical assumptions.
in this case, the outhole relay powers when a ball closes the outhole switch, and stays powered until score motor 6B switch opens (remember from above that will happen after 1A closes/opens). Since powering the outhole relay runs the score motor, you run into wtf #2. How can a ball be sitting in the outhole to power the outhole relay, maybe get kicked to the shooter and score points before 6B opens?
it can't, so either the outhole relay has to hold itself powered until points are scored, or the ball count unit doesn't step up until the ball drains.
you can go find that stuff on the schem, and the lamp part of the schematic gives a big hint ... the ball in play 1 light is on when the unit is reset, so on this game ball in play steps when the ball for the last enabled player drains AND they've scored points without getting an extra ball.
note some games would reset to have no ball in play lights lit, and the unit would step up once when points were scored, the ball was kicked to the shooter, or ball was shot. You always want to use the schematic for the game, and if you only have a schem for a similar game, be skeptical.