I use SimpleScreenRecorder. You can choose "Separate file per segment", and each time you pause you get a new file.

I then delete the mistaken takes, and concatenate everything with an FFMPEG script.

On Mac: Quicktime + iMovie.

Quicktime has a very simple/flexible screen recorder built in. You can record a section of your screen, or full screen. If you make a mistake, pause for 5 seconds and then just do a re-take on the fly.

Import the QT recording into iMovie. Clip out of the unwanted sections and export your video.

We'll let you download the video after, and delete it too! Maybe all in one-button. I'm assuming you don't want cloud because of your data? Even better for us, we don't have to pay for hosting..