So, for some reason, the 'select external storage' option suddenly and randomly appeared. Out of nowhere, and hours after I had started up and neglected the update prompts for the so manieth time. I have NO idea why or how this happened.
I finally got the chance to select external storage, but the download still wouldn't start. And then I had to wait for this screen to randomly pop-up again, which it eventually did.
Anyway, it turns out that even when using external storage, there still has to be some free space on the C disk. I had to clear up another 2 GB instead of 8 GB now, which was still a challenge, but after literally deinstallling all removable software (which
wasn't much, considering most of it already is on my external storage.. duh) I finally had cleared enough space for this update.
After the update the space I had cleared up was free again... so I could reinstall the stuff I removed but I haven't bothered yet.
Good luck to everyone who's still left in the dark.