Using Quicktime to record a video from the web on Macbook Pro

Make sure that you are not in full screen mode and then you can easily select another window. Move and arrange app windows on Mac - Apple Support

One example if why you would record your screen is if you were creating a software tutorial or trying to demonstrate something.

Apple provides very clear instructions for two ways to do this: How to record the screen on your Mac - Apple Support

I just gave it a try and it works as advertised. Using your example:

1. Launch Photos, find the video and double-click it to get ready
2. Launch QuickTime Player (click Cancel in the file selection dialog box if one opened)
3. Select File > New Screen Recording
4. At the bottom of the screen you will see the toolbar, select the "Record selected portion" button
5. Use your mouse to move and resize the highlighted selection area over the part of photos that you want to record
6. Click "Record"
7. Click the play triangle if playing a video, or scroll around in photos and show a few things as an example
8. When done, click the Stop button that will be in your Mac's menu bar (upper right)
9. The resulting recording will be saved to your Desktop (unless changed in Options before recording)

Depending on your experience level, I'm guessing that what you really want to do can be accomplished best using the command line tool "yt-dlp" (https://ytdl-org.github.io/youtube-dl/). You can install that tool using Homebrew (https://brew.sh/). Learning to use it for your intended use is up to you.

As FishingAddict points out, it is better to not re-record, but get the source file directly. For many sites this can be done, but not all (some content providers protect their content from this). So it depends on which site you would like to capture from.