Is there a way to set another shortcut for "Eject" in the Finder without using a third-party application?
The only way I can think of to achieve this without any 3rd party software is to do the following steps:


Use Automator to create a service that runs an Apple Script
Use System Preferences to assign a keyboard shortcut to the service


Below are the steps involved for each. Don't worry, the whole thing should only take 5-10 mins.

1. Create a service using Automator


Launch Automator (usually found within your Applications folder)
Go to File > New.
Select Service and click Choose.
In the top right hand of the window that appears, ensure that "No input" is selected from the Service receives drop-down list.
Ensure the In drop-down list is showing "Any application".
In the second column containing the long list, find "Run AppleScript" and double-click on it.
This will add the Run AppleScript window on the right.
Now, replace the ( Your script goes here ) with the following code:


set exceptionsList to {"HD1", "HD2", "HD3", "HD4"}
tell application "Finder"
    set diskList to the disks
    repeat with mountedDisk in diskList
        if name of mountedDisk is not in exceptionsList then
            eject mountedDisk
        end if
    end repeat
end tell


In the first line of code, add your exceptions by changing the references to HD1, HD2, etc to any names of drives you do not want to eject. For example, HD1 should be replaced by your Macintosh HD (or whatever it's called). You may also want to add any Time Machine drives to the list, etc. You can add more by adding a comma and placing the name within quotes. Likewise, you can remove any you don't need.
Save the service using a meaningful name (e.g. EjectAll).


Now to the next step.

2. Creating your shortcut


Go to System Preferences > Keyboard > Shortcuts.
Select Services from the sidebar.
Find your service (it'll be in the list on the right-hand side).
Add a shortcut by double clicking on the service name.
Now go to System Preferences > Security &amp; Privacy > Privacy.
Select Accessibility in the sidebar.
Click on the + sign (you may need to unlock the padlock if it's locked).
Add Automator.
Add Finder (to find this you will need to navigate to /System/Library/CoreServices/Finder.app).
Exit your System Preferences.


Now you should be able to do two things:

1. Eject your drives from any application by going to the Services list within any Application menu (e.g. Finder > Services, Safari > Services, Mail > Services, etc) and select the service you just created. This will eject all drives not within your exception list.

2. Use the keyboard shortcut to run the service.

NOTE: I am aware that for some users keyboard shortcuts do not seem to work even though they've been assigned to a service. This is a whole other topic, but if this happens in your case, at least you should now be able to eject all drives from within any application you're using.

Hope this helps you and others.