i use the English term "screen recorders" to designate the whole family of intelligent video authoring software which record the stuff displayed on the screen. Many youtube tutorial videos are the product of intelligent screen recorders, some of which deploy presentations or slides (techniques) when the screen contents dont change much over time.
For example if you record a displayed static GIF-image (50kB in size) for 1 hour as a video clip (with no background sound), the resulting *.AVI (or whatever optimal video format should be used in this recording task) should not be much bigger than the 50kB .. if you have chosen the perfect video format, video settings, recording settings, video recording software, etc.
I am not an expert at screen recording (and screen recording software products) but obviously Kelby really knows how to produce vivid videos (you see live people, human beings, and screenshot-type video sections where Adobe products are featured).
In the end your problem comes down to "authoring a video file" and "creating video contents", and that has nothing to do with "saving an original internet video stream" (produced by the website's owner).
The latter is much easier (and fun). That's all i can do.
