How to make a bootable, *usable* macOS USB key (NOT an installer)

The way to do this is to create multiple partitions on the USB drive. There is a Tech Republic article that gives the full details on how to do this:

http://www.techrepublic.com/article/pro-tip-create-an-os-x-multiboot-usb-install-drive/

In summary, TechRepublic gives the steps:

1. Using Disk Utility, create separate partitions for each version of MacOS that will be booted. (Mac OS Extended (Journaled) format; In "Options" choose "GUID Partition Table.") It's a good idea to name the partitions for the Mac OS version you'll be booting.

2. Copy the relevant contents to the particular partitions. This is different for different versions of MacOS:

   • Apple OS X 10.7 (Lion), 10.8 (Mountain Lion):
     a. Right-click the installer and select Show Package Contents from the context menu;
     b. Drill-down through the Contents | Shared Support directories and drag-drop the InstallESD.dmg to the desktop;
     c. Using Disk Utility's "Restore" tab, Perform a Restore into the appropriate partition, using InstallESD.dmg as the source. This takes about 30 minutes.

   • Apple OS X 10.4 (Tiger), 10.5 (Leopard), 10.6 (Snow Leopard):
     Since these were distributed as DVDs, you restore from the DVD or ISO image. (DVD restore will be much slower than the file restore method.) Use Disk Utility's "Restore" tab with the DVD or ISO image as the source.

   • Apple OS X 10.9 (Mavericks) and 10.10 (Yosemite):
     Ise the new "createinstallmedia" command. If your installer.app is not in /Applications, modify the "-applicationpath" argument appropriately.)
     For example:

     sudo /Applications/Install\ OS\ X\ Yosemite.app/Contents/Resources/createinstallmedia —volume /Volumes/DRIVE_LABEL —applicationpath /Applications/Install\ OS\ X\ Yosemite.app —nointeraction
     

I expect that the method for 10.7 and 10.8 will work for later versions of MacOS as well.

No, you can't just copy the files from one to another drive and assume that the new drive will install El Capitan. There is a special command to create such an install drive, called createinstallmedia which runs in Terminal

1. Download the OS X installer from the Mac App Store. Quit the installer if it opens automatically after downloading. The installer will be in your Applications folder.
2. Mount your USB flash drive or other volume. You could also use a secondary internal partition.
3. Open the Terminal app, which is in the Utilities folder of your Applications folder.
4. Use the createinstallmedia command in Terminal to create the bootable installer.

If you're creating from an existing El Capitan machine, please use the following command.

sudo /Applications/Install\ OS\ X\ El\ Capitan.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume --applicationpath /Applications/Install\ OS\ X\ El\ Capitan.app

I haven't personally tried doing this with an external hard drive instead of a thumb drive. But I don't see any reason for it not to work.

For more details and how to configure the above command (it depends on the OS you're installing to the USB drive from), please refer to https://support.apple.com/en-gb/HT201372