If you were to achieve the "left mouse button is down" while not achieving "mouse is above the menu bar" and the left mouse button went down on the menu bar, then you are experiencing a drag.
What you have described may be the start of a drag. On Android, if the mouse button is held down long enough and there is no movement then it will register as a longpress.
So in the modern world of UI and UX, these common event names may be suitable for the display of the menu:
onmousedown (web), onpointerdown (web), onLongpress (Android)
While these common event names may be suitable for the hiding of the menu:
onmouseout (web), onmouseleave (web), onlostfocus (web), onBlur (Android)
As for a name, it is like an archaic, needy hover dropdown menu. Why not give it a name and see if it sticks? "Needy Hover Menu"?