In Gnome Shell you can press Control+Shift+Alt+R to start and stop the built-in screencast tool. What is the name of this tool? Is it possible to invoke it from within Unity?
The gnome-shell screencast tool you are referring to is built inside the shell and you cannot use it unless you are using GNOME session. It is part of the session and you can liken it to being something like Unity's top panel, which doesn't come outside of Unity.
And, there is no specific name given to this tool other than the generic built-in GNOME screencast tool.
If you are interested in creating screencasts, you can follow this question for alternative tools:
