Since this is not explicitly mentioned in the rules (or the FAQ):
1. When production is triggered for the first time in a round:
what happens to the production chips afterwards. Do they stay on the production fields? YES, THEY DO.
2. When production is triggered for the second time in a round:
Can I only use the free production fields (without a chip on them)? YES, THIS IS THE CASE
We played that the production chips stay on the fields and the second time you can only use the remaining 2 free production fields.
Seemed to make the most sense, but would like to hear if this is correct.
SO YES, YOU PLAYED IT CORRECTLY.
PS: NOT EVERYTHING IS IN THE RULES, WHEN WE HAVEN'T SAID THAT THE CHIPS HAVE TO BE REMOVED, THAN IT MAKES SENSE NOT TO DO SO:) BUT NO PROBLEM, GLAD TO HAVE BEEN ABLE TO CLARIFY;)