I would not concentrate on "where did I do what - and (re)check every place" --- I wold say "I bought a non-running pin - can I fix it ?".
In post-1 You write: "I push the Start-button, the ONLY thing happening is: The Score-Motor turns about 2 to 3 times --- NOTHING else". This cannot be true - The Strat-Button does NOT directly start the Score-Motor - most likely the "S-Start-Relay" is actuated - and by a closing switch (on s-Relay" -> Score-Motor starts and make turns. Question: "A TURN" ? "A REVOLUTION" ?
My definition (I am German speaking) - I hope its everybodys definition: The Turntable on Gottlieb Score-Motors is divided by/into (?) THREE. (For me) ONE TURN is "turning 120 degrees" of the turntable - therefore 1/3 revolution.
So does Your score-Motor turn 2-3 revolutions - or 2/3 to 1 revolution ?
It is a Gottlieb Pin - the travel of the Switch-Blades on the relays is short - it is a small gap (when a switch is open) - You must have a good look. The anchor-plate of a relay MUST sit in proper place.
The very first relay to investigate (to me) is the S-Start-Relay.
(Have some points on the score-Reels - to see if the pin does reset the Score-Reels) - does the S-Start-Relay actuate when You push the Start-Button ? If NOT: What happens when You actuate the armature (anchor plate) by hand ? (I would use a wooden stick - a ruler from school is fine - I would NOT use a screwdriver, reason: Safety reason, insulated wooden stick ...)
I look in the schema of my Orbit -> the S-Relay has many switches - it actuates (should actuate) the Z1-, Z2-, Z3- 1,2,3,3 Player-RESET-Relays, the SB-Reset-Control-Relay ...
So have a close look at the S-Relay and the switches, blades mounted - does the plate sits good, returning spring mounted - blades straight or bent, wires soldered-on (bent in the solder-lug-side) ?
Question for "mabe in the future needed": Is there a stripe of paper with text - telling us "from left to right - the relays in the huge Relay Bank are (example) some relay, 2nd Player-Relay, 3rd-, 4th-Pl-Relay, 1st ball relay, Game-Over-Relay etc."
(((Working on Gottliebs - sometimes it helps - before starting a new game: Manually reset the Relay Bank -> the let drop the 1st ball relay, let drop Last ball relay and the Game-Over-Relay)))
P.S.: The Relay-Bank: http://www.ipdb.org/showpic.pl?id=1332&picno=26197 - to the left, above the chimes.
P.P.S.: Here http://www.ipdb.org/showpic.pl?id=1332&picno=26198 -> almost on top - way to the right -> the socalled "Through-Switch" - when a new ball is kicked over to the shooting alley -> the ball rolls over thes switch -> switch does close -> ACTUATES THE P-ADD-PLAYER-UNIT-RELAY -> in the Backbox -> next player or next ball...