I agree with the mayors. The name, "Evergreen Line", is a brand that needs to remain. It should not be removed just because Translink feels like it.
When Millenium Line came to service, they didn't remove the "Expo Line" on the portions where the two lines overlapped. The "Millenium Line" and "Expo Line" is used to identify which route the trains are traversing up to their final destination.
So it only makes sense that "Evergreen Line" is used for the route that a trainset is using if it were to travel from Douglas / LaFarge to whatever end point it will travel, whether its Lougheed, Commercial, VCC/Clarke or eventually Arbutus (and only up to Arbutus forever

).
If the route of the current Millenium Line, which travels from Waterfront to VCC/Clarke gets shortened up to Lougheed or Production, then so shall it remain that way. There is no point rebranding or rerouting "Millenium Line" to include Douglas / LaFarge, Coquitlam Central and so on as per what Translink is mulling.
So on 2016 (assuming Millenium ends at Production Way), the robot announcer would say this...
Coming from Waterfront:
"Expo line to King George"
"Millenium Line to Production Way"
Coming from Douglas / LaFarge:
"Evergreen Line to VCC / Clarke"