Specifically gems but also specific items like BFA's Midnight Salmon. I have the "auto-deposit" turned off and every time I click on "Deposit All Reagents" in the bank they are deposited along with other stuff since they are, indeed, "reagents", but I want to keep them in my bags instead.

I believe that button just follows Blizzard's built-in function, which directly deposits everything that can go into the Reagent Bank. You're probably better off using something like TSM's BankUI module, which allows you to create rules for this.

(Edit: Misread "bags" as "bank.")