Replace internal SSD (faulty) with a new, which requires "macOS 10.13 High Sierra"

Did this laptop ever have macOS 10.13+ installed at any time in the past? If not, then I'm not sure you can do so now without a working original Apple OEM SSD installed internally as that is required for the system firmware update included with the macOS 10.13+ installer.

As far as the alert message "this update requires macos version 10.13", it appears you downloaded a macOS 10.13 update patch and not the full installer. The macOS 10.13 full installer needs to be downloaded from the App Store. You can use the links in this Apple article to get the macOS 10.13 installer as well as the optional instructions for creating a bootable macOS USB installer.

How to create a bootable installer for macOS - Apple Support

Like I said though, I don't think you will be able to install macOS 10.13 even to an external drive without a working Apple OEM SSD installed internally because the macOS installer requires it for the system firmware update. The laptop requires a system firmware update only available from a macOS 10.13+ installer before the third party NVMe SSD will be visible to macOS 10.13+. If macOS 10.13+ was already installed on this laptop at some point in the past, then booting from a macOS 10.13+ installer should work. If the SSD is not seen, then it may mean the SSD adapter is not compatible (Sintech is the only adapter that seems to work well), or the SSD is bad or not compatible.

In my own personal experience repairing our organization's computers, the SSD is the weakest link and more likely to fail than the Logic Board. If you are using a third party M.2 SSD with an SSD adapter, then the SSD adapter may be the problem. From reading these forums it appears only the Sintech adapter is recommended by users using a third party M.2 SSD. Of course this assumes the Logic Board is otherwise healthy.

Unfortunately getting a compatible original Apple OEM SSD is not that simple. You need to make sure the SSD you get is made for your exact model laptop (or known to be compatible). While other Apple OEM SSDs from other models may fit into the SSD connector, macOS may not work with them if the SSD was made for another model Mac (sometimes macOS will work fine, but other times it won't). I personally encountered this myself. I had an original Apple OEM SSD which macOS would not even boot to the macOS installer, but worked perfectly fine using a non-Apple OS.

Once you have installed macOS 10.13+ to the laptop, the system firmware should be updated to work with an NVMe SSD. If you now installed the other third party NVMe SSD, then you should now be able to see it with a macOS 10.13+ installer. If you are using an M.2 SSD you would need to use an SSD adapter which can affect compatibility. The Sintech adapter seems to be the one people on these forums reports works the best. Of course, OWC offers an NVMe SSD with Apple's proprietary SSD connector so that no adapter is needed (it also means there are no external enclosures for the OWC SSD to access the data if the laptop has a hardware failure).

Glad you were able find an original Apple SSD to get macOS installed.