First of all, Windows (the version being used in the running of the Recovery Media), under a WinPE BOOTable RAM image (almost all Recovery Media) does not use "C:", it uses "X".
My guess on Klaver7's problem is that his Recovery Media does not include the driver necessary for that storage device type to show up using the Recovery Media. As such, we need to know a few things. Your 1st image shows the LIVE System device to be a Western Digital NvME Solid State disk. So now we need to know... how was your recovery media built (were you using the Microsoft download option <WinPE> or not <the resident WinRE would be used>.
Depending on your method, the required driver for the Recovery Media seems to be missing or you would see that device in the 3rd image. I would try rebuilding the Recovery Media using the option not used for the original attempt. A Recovery Media built using the resident WinRE should have that driver in its residency. A Media built with the WinPE option (Microsoft download + Automatic Driver injection) should also have that driver in residence.
If neither work, there is an issue with the needed driver for that System device.
My guess on Klaver7's problem is that his Recovery Media does not include the driver necessary for that storage device type to show up using the Recovery Media. As such, we need to know a few things. Your 1st image shows the LIVE System device to be a Western Digital NvME Solid State disk. So now we need to know... how was your recovery media built (were you using the Microsoft download option <WinPE> or not <the resident WinRE would be used>.
Depending on your method, the required driver for the Recovery Media seems to be missing or you would see that device in the 3rd image. I would try rebuilding the Recovery Media using the option not used for the original attempt. A Recovery Media built using the resident WinRE should have that driver in its residency. A Media built with the WinPE option (Microsoft download + Automatic Driver injection) should also have that driver in residence.
If neither work, there is an issue with the needed driver for that System device.