Unable to Update after initial installation

[Şerban S.]

@Serban
Nice helpful answers in the forums, I clicked your Thank button. I was first! )

Thanks! I returned the favor!

[...] Could it be that the USB key (i.e. the "Live" OS) is still the one booting after installation

I seriously doubt it.
Supposing we forget to get out the stick. What update is it running?
This would be possible only IF the Update command was issued FROM WITHIN the llive distribution.
While this is definitely possible, since we are on an "unwritable" medium, the ISO having a fixed size, I still doubt this happened.
I was once curious to see what happens if I trigger the update from a live distribution and the console message usually points to the media that is out of space.
Since the snippet contains only strictly the messages related to wget and the errors that occured, it is impossible to know for sure what really happened, so I went on the idea of a partition that is full since I would never issue an update command on a medium that is "ReadOnly", as the Live USB distros look like for the system. Due to the fixed size of the ISOs, even if there is lots of space on the drive, it remains unpartitioned, thus it looks like "Full". Knowing that, why would one issue "Update"?
But, as you pointed out, IT IS POSSIBLE.

[...] and/or that the installation was done on the wrong disk? [...]

Unlikely. It's waaaaay too hard to pick unintentionally EXACTLY a partition that is some 1 - 2 MB only larger than the installed OS. That would be wizardry! I mean, do you know exactly +/- 1 MB what space would it occupy after installation and size the partition exactly at this size UNINTENTIONALLY?
First scenario, is more likely to happen (Issue Update from the Live distro) than targeting a drive that has exactely the size of the installed distro and nothing more.
As far as I know, the script generated by wget (the update lists), is at most 50 - 100 KB (many times, even less!) so picking by mistake a partition with the exact size of the installed OS +/- 10 KB is really a weird scenario.
It IS possible, only on an live distro, though. This is exactely the case: fixed size of the partition.

In order to use "Persistent", you need a special ISO, with a special ISO Installer script to (A) check the amount of free space on the drive then ask you if you want it to partition and format as "Persistent" and (B), a special installation script that knows that the live distro actually CAN write on the stick and where exactely to do this.
I lack the knowledge if Linux Lite Live can do this. As far as I know, usually this kind of magic, is done for specific distros that are intentionally built to run from an USB stick, and they are usually "minimalist" distros, with a lot of tweaking for the OS, since running from USB is very slow and this requires high level knowledge on optimizing the kernel and the other components to use the lowest possible amount of RAM and CPU. Most of those distros, lack tons of features, exactely due to the resources problem.
I know this, because I tested myself the USB installation of some distros and they run like on a 386 CPU, regardless the amount of RAM or number of threads (nucleuses) per CPU.
Anyone can in fact test the speed of an USB stick, using the speed test from within hardinfo. They are comparable with the speed of a Blue-Ray, maybe a little faster sometimes.

All the best!