Only the first disk shown in your pictures, (internal), is "real". (synthesized) is how your Mac will see the real drive once it's complete, but you can ignore this one for now. All the others (disk image) are made by Internet Recovery as temporary structures, part of your recovery install & will vanish once you are booted normally.
What looks to have happened is you have formatted your internal drive, but not added a volume, only the APFS container.
The 'quick fix' would seem to be to format it again from Disk Utility. Set the View menu to 'Show all devices' first, then select your actual physical drive, not anything within it [you will see devices & their contents as a set of hierarchical folders in this View, so choose the top level] & use the 'Erase' function.
It should then make a new EFI and a new Container Disk, containing one Volume, named as you set in the previous dialog. That should now be ready to accept your install.