My main question is this:

When you resolve the second option, to persuade the synth, what happens to the synth? Thematically it must surely be removed as an enemy, yet I don’t see any instruction to remove it. Am I missing something?

Secondary question:

How would you trigger option 1, placing the third Shield token on the map?

Firstly, the third shield token is placed in one of the results of card 017 which was added to the Wasteland encounter deck when card 016 was staged. This is a common thing with this game were a card that is staged doesn't really make sense until you find the card that was added along with it.

Also, there is a confusion that people not familiar with Fallout 4 often make. The Synth enemies that are represented by the Shield tokens are NOT the same as the synths often talked about in the cards. The enemies represent the Gen-1 robotic synths whereas the synths talked about on the cards are the human-like Gen-3 synths.

Finally, in this particular situation, the synths that you are persuading are found later in cards 020 and 134 that are added as a result of picking option #2.

1 hour ago, Hedgehobbit said:

Also, there is a confusion that people not familiar with Fallout 4 often make. The Synth enemies that are represented by the Shield tokens are NOT the same as the synths often talked about in the cards.

Butbutbut, I have to be adjacent to a Shield token to do the action, so surely it is meant to be that I'm persuading that enemy synth to become a nice synth? Maybe you're persuading one of the non-passing synths to turn good, or maybe some of the enemies are human-like synths.