Actual working tutorial of installing catalina on an external drive?
Plug the external drive to your MacBook and Launch Disk Utility.

Select the external drive on the left-hand side panel, select Edit&gt;Erase, choose APFS and GUID partition map and press Erase:


Download Catalina installer from your genuine MacBook App Store by clicking this link.

After downloading, the installer launches automatically. If you want to install Catalina at a later time, you can find the installer in the Applications folder:


Follow the onscreen instructions, and when asked where to install Catalina, choose the external disk.

You MacBook will restart and proceed with the installation.


When the installation is over, to boot from the external disk, start up your Mac and press and hold the Option key to display the Startup Manager. Choose the external disk and click the arrow under its icon.
See these support articles about installing Catalina and erasing a disk for more information.