The blue spinning indicate that the computer is busy with running a process in the background. and there is no specific name to it.
What is the spinning blue circle called?
A throbber, also known as a loading icon, is an animated graphical control element used to show that a computer program is performing an action in the background (such as downloading content, conducting intensive calculations or communicating with an external device).
What is the spinning colored wheel on a Mac?
What is the Mac spinning wheel? This rainbow-colored spinning wheel appears whenever an app on your Mac needs more time to perform a particular task and becomes unresponsive while doing it. Each app on your Mac has what’s known as a window server, a system process that helps an app communicate with your screen.
