"Root" store solved it

The solution was to install it into the Root store, which puts it under Registry (if you view physical stores in the Certificates snap-in), instead of the AuthRoot store, which puts it under Third-Party. I don't know why it was disappearing from Third-Party, but it hasn't happened since I made this change.

Thanks to Charlie Wilson for pointing me in the right direction. I just wanted to type out the full answer in case anyone else had this problem.

One can edit the certificates on the local computer this way:

1. Start → Run: mmc.exe
2. Menu: File → Add/Remove Snap-in…
3. Under Available snap-ins, select Certificates and press Add.
4. Select Computer Account for the certificates to manage. Press Next.
5. Select Local Computer and press Finish.
6. Press OK to return to the management console.

Once the local certificates are displayed:

1. Click to open "Trusted root Certificate Authorities"
2. Right-click on Certificates → All tasks → Import
3. Browse to your certificate to import it.

You can save this console for future use via the menu File → Save As.