Yesterday I downloaded Neon from its official download page, which was version 15.8.2. I wrote the ISO to my flash drive with Rufus and booted it from the boot menu (esc during the motherboard screen).

That version had a bug that caused the user installer to crash at 66% while doing something with Grub. I then formatted my flash drive, downloaded an ISO from December that someone gave me so I could install that and update to 15.8.2 afterwards as someone suggested, still got the error. Then a new version, called "18.04" in the info file, came out very shortly after. I wrote that to the USB with Rufus again, booted it, and got the same error, but then I noticed something: system information said it was still 15.8.2, the broken version.

Why isn't the version I'm booting getting recognised, how is the older one still there? It should be noted I'm attempting to make it a dual boot with Windows, and I'm allocating my ram in gigabytes (16) for the swap area in my 40GB of free space, then allocating the rest as ext4 with just "/" as the mount point.

18.04 is referring to the version of Ubuntu that Neon is built on top of.
The 5.18.x is the version of Plasma. This displays during boot, and changes when the Plasma version changes.
Technically, I do not believe that there is an actual 'version' of KDE Neon, the ISO image is rebuilt regularly, and is named by the date.

I have no idea what or where you are seeing the system information, but my guess is that the USB stick was given a label or name when you first created the installer, and Windows saw that, and remembered it. You created a new installer from the older ISO, but Windows is seeing it as the same stick, and is showing the old label. So, you can probably ignore anything you see, except from what is presented to you by the live installer.

I'd suggest maybe trying a different tool, such as Balena Etcher to make the bootable installer, or verify that the settings in Rufus are correct.
If you have not already done so, make sure you have an internet connection during the install. I have seen this cause problems with grub on various Ubuntu-type OS installs every now and again over the years. It is trying to download a needed file, and fails sometimes.

Balena Etcher did the trick, thank you so much. I don't know what was up with Rufus.