Update 2024-08-08: 1.0.9157 works too.

On your Windows 7 PC

Step 1: Install 64-bit Discord 1.0.9034

This is the latest version of Discord to run on Windows 7 officially, and what the Windows 7/8.1 installer had, albeit 32-bit.
You can download it from here or the Download Center.
After that, just install it.

_{Honestly, you could stop here, 1.0.9034 should keep working for the forseeable future}

On your Windows 10 PC

Step 2: Install the latest 64-bit Discord version

You can download it from here
At the time of writing, it is version 1.0.9154.
Just install it on your Windows 10 PC.

Step 3: Copy the app-1.0.xxxx folder to your Windows 7 PC

This folder stores all of Discord's actual app data.
It's stored in %LOCALAPPDATA%\Discord
Just copy it as a whole to your Windows 7 PC.

On your Windows 7 PC

Step 4: Replace Discord 1.0.9034 with whatever the latest is

Go to %LOCALAPPDATA%\Discord
Do not delete app-1.0.9034, doing so will cause the updater to redownload it.
Go into the app-1.0.9034 folder, and empty it out.
Copy in the contents of the app-1.0.xxxx folder from the Windows 10 PC that you got with the newer version of Discord

Step 5: Make it work

Now, to make it all work, edit Discord.exe with CFF Explorer and change the subsystem version to 6.1.

Then, go up a folder and enable VxKex on Update.exe
Now, just run it.

Enjoy!