Reading the description of the edge in the book, I think you have it right the first time.
The player takes it once when the character becomes a department head. Taking it again would upgrade it to level 2 indicating a captaincy or equivalent.
Note that edges do not have "tiers" as such (professional abilities do) just levels that may or may not be upgradable. Each time you take an edge upgrades it to a new level - they do not need "maintenance" as such!
Also advancements represent packages of improvements, not periods of time as you seem to be thinking. You do not need to account for everything you did, just what you got out of it!
Jon
"There are worlds out there where the sky is burning, where the sea is asleep and the rivers dream; people made of smoke and cities made of song.
Somewhere there's danger, somewhere there's injustice, and somewhere else the tea is getting cold. Come on, Ace, we've got work to do."
THE DOCTOR, "Survival" (Doctor Who)