You can think of a brigade as a band of people who are organised to act together: they have roles, they train, and when a call comes they move as one body. Put fire in front of it and the picture is fixed on one job — going towards the smoke while everyone else goes away from it. Both meanings of fire brigade are that same picture; what changes is who the people are and whether this is their paid work.
Two branches
公家的消防隊
這一支是把核心畫面制度化以後的樣子：整個城市養一支隊伍，全職、輪班、隨時待命。
noun · B2
The public service whose job is to put out fires and to help when there is smoke, a trapped person, or another emergency. This is the usual British word for it; American English usually says fire department.
消防隊
自己人組起來的救火隊
一樣是編好隊、一起行動，只是把「全職」那一層拿掉：人本來就在這裡上班或生活，救火是額外扛的任務。
noun · B2
A group of people who train together to fight fires where they live or work, but who are not paid firefighters. In a company, a factory, or a ship, the members do this on top of their normal job.
義消隊、廠內自衛消防隊