We have adopted that "the army, internal security, industry, and international relations" are independent institutions (ajhizah) and not departments (dawa'ir), so why do we use the word 'department' (da'irah) and not 'institution' (jihaz)? May Allah reward you.
Yes, each one of them is an independent institution (jihaz), but it is an administrative institution (jihaz idari) that may be named as such. However it is not related to the institution of people's affairs (jihaz masalih) rather it is an independent institution (jihaz).
Refer to the chapter: "Bayt al-Mal" State Treasury (p. 135 Arabic ed. / p. 129 English ed. The Institutions of State in the Khilafah), it states the following:
"We have already adopted that the wali is given a special authority that excludes the army, judiciary and funds. Thus, the whole army will be a central institution (presided over by Amir al-Jihad). The judiciary will be a central institution (known as judiciary), and the entire funds form a central institution (known as bayt al-mal), which is separate from any other institution in the State, and follows the Khalifah as do the other institutions."
As you can see, it is called an institution even though it is an administrative department. However it is a centralized department (da'irah markaziyyah) i.e. not affiliated with the Administration department (people's affairs). Likewise, "the whole army will be a central department (presided over by Amir al-Jihad)," "The judiciary will be a central department," and "funds form a central department (known as bayt al-mal)."
Therefore each institution (jihaz) whose work includes administration is called a department, and if there is the fear of confusion which department is considered from the institutions (ajhizah), it is possible to add the word 'central'. If there is no fear of confusion, then department is appropriate, and it is clear that the army, internal security, industry, and international relations are not confused with the administrative department (People's Affairs) in which each department is called a department.
21 Dhul Hijjah 1433 AH