Have I got this right:
Every turn, the specialists in front of you score you points equal to the building markers of their colour.
However, for every other purpose (end game trigger, Project Cards (Single Specialist & Double Specialist), then the value we are concerned about is the Specialist Marker (the round one).
Almost 100% yes. Yes, your specialists score at the end of your turns based on the BUILDING value of the same color as the specialist. For end game trigger, three specialist values at 5+ will do it. Of the four types of Project cards, single and double specialists both are based on specialist value, but you can use the BUILDING value to assess the number of buildings of a certain type renewed.
I hope that helps!