Always show function keys on control strip when using terminal
The reason for this is that Apple doesn't seem to treat its Terminal app in the same way as other apps in terms of permanently setting Functions keys in the Touch Bar. Instead, you need to follow these steps:


Launch the Terminal app
Go to View > Customize Touch Bar
Drag the Function keys onto the Touch Bar
Click Done


Another option is to use another app as your terminal (such as iTerm2) and follow the usual steps for permanently setting the Function keys on the Touch Bar.

Based on your screenshots, it seems you know the steps for doing this, but just to be sure and for the benefit of others:


Go to Apple > System Preferences
Select the Keyboard preference pane
Click on the Shortcuts tab
In the left sidebar, select the Function Keys option
On the right-hand side click on the plus + button
Search for the app you want to use (e.g. iTerm2)
Add the app


Now, whenever you are using the app, the Touch Bar should display all the function keys.