What are the flavors of all those "systems":系统,体系,制度
装置 'device' doesn't really belong here. Here are some definitions from the 教育部國語辭典:
系统: 同類事物，按一定秩序相連屬，而自成一整體
"objects/affairs belonging to the same category, joined together in a specific order to form a unity." 
The emphasis is on order, thus can be used to form adjectivals such as 很有系統 "very systematic"
體系: 由許多要素構成，具有一定條理組合成的整體。如：「哲學體系」
"a unity formed from numerous elements which have a connecting principle, such as 'philosophical system.'" The emphasis is on the underlying unity of the elements, hence principles.
制度:經制定而為大家共同遵守認同的辦事準則
"a code for the conduct of affairs established by common understanding and agreement." The emphasis is on how things are done, hence this is commonly used with political affairs.
體制: 一定的規則、制度. "fixed rules; a system of operation." The emphasis is on the set nature of the system. This can also be used for written formats.