Thank you all for your tips and help!
Long story short… I backed up my files and committed to wiping the phone, so I was thankfully relaxed about things not working straight away. I’m on Linux (Debian) and tried this how to guide first – that resulted in the error message fastboot: error: cannot load '/home/anna/Downloads/FP3/tz.img': No such file or directory.
That made sense to me because the file wasn’t there; I had however checked for file integrity after the download so I thought that something else must be wrong. (I just thought to check whether I had unpacked everything properly, and I HADN’T – so the missing file was actually in the zip. D’oh. 
)
Next I tried the snap easy installer, which got hung up on detecting my phone’s fastboot mode in step 2. Others have had that problem, but only under Windows, so I was stumped.
End of story, I did a factory reset, and after that I had ALL the updates available in the updater, including the latest 2.0-something version. I downloaded that and installed it successfully, and after a reboot it said my OS was corrupt and the only option was another factory reset.
And now after another factory reset it just booted into version 2.0, my internet connection is working perfectly again, only 22% of disk space are in use, and I will now spend so much time reinstalling apps. But I am very happy 
Thanks again, I wish everybody who comes across this post good luck with their updates!