In the mod list menu, the icon of the new version is constantly blinking.

It's say you have the version "1.0.1" and the latest is "1.0.1.95-alpha".

Are you sure to have the latest version ?

I install all the mods through the Twitch launcher:

Your mod is not updated for some reason through the Twitch launcher?

Maybe the update is not automatic, you have to update it your self.
The latest is available on curse forge and on the twitch launcher

I specifically went into the mods folder and copied the file name: CraftStudioAPI-universal-1.0.1.95-mc1.12-alpha.jar