I'm looking for the correct terminology for the little triangle next to the task that will display the subtask.
I would call it an "Expander", based on the functionality it provides. Although in this case it is used for Tasks, that particular type of UI control exists in many other UIs, including the Windows Explorer:
In that case, and in many others, the functionality it provides is to expand a view of child/sub items.
I have no idea what the official Microsoft name for that control is.
In fact , I don't know if this arrow has a specific name or not ? ,but usually, I tried to call something based on its functionality or what it denote to ?.
So This arrow denotes to the
multi-level parent task that can
collapse as your first image to hide related sub-tasks or
expanded as the second image to show related subtasks.
Also to build your multi-level task structure in Task List, you can use the above ribbon by using
indent (move the task down level),
outdent (move the task up level) option
So if you can't find an official name for this arrow, you can call it based on its functionality as
Task Grouping arrow or
Multi-Level task arrow or
Parent task arrow !