Some comments on the shown pictures.
That's not about the failing auto-install
That's about the 2nd part where you have tried a custom-installation on existing partitions with EFI-mode.
On custom-installation you need to make sure to have prepared the partitions for EFI-installs:
On a whole disk install prepare drive this way:
1. Create a new partiton table (Gparted menu -> Device ) and do choose "gpt"
2. Create an ESP-partition
The shown /dev/nvme0n1p1 is a 256MB vfat partition - but it is not a ESP (EFI SystemPartion, yet)
Within Gparted you need to "mark" the /dev/nvme0n1p1 as "esp" + "boot" , otherwise this partition is not recognised as ESP.
3. If not install encrypted, I do recommend to not choose a extra boot-partition - just leave boot as it is, so it will be part of the "/"-root partition. There is no advantage to have extra boot-partiton for non-encrypted installs, but the opposite, it make recovery, e.g. chroot into more complex. Also MX-boot repair might not work with extra boot, due to difficulties with chroot into extra-boot installs.
4. You might now wonder why the installer proceeded with the installation which had no ESP (EFI-partition prepared) on the nvme0n1 drive. The answer is: Within the system is an ESP-partition, which is on the LiveUSB.
For the UEFI-system it doesn't matter which ESP you boot from. So you could in theory place the EFI-grub loader (which is part of the grub-install ) onto the USBstick to boot into an installed system on the internal drive.
So choosing MBR during an EFI install makes not really much sense. Again in theory you can install the legacy GRUB-loader into MBR and later boot in legacy mode from the drive. But on GPT drives you need a extra "bios-grub" partition to hold the legace MBR-Grub-bootloader. Yes in this way I sometimes prepare my system to be bootable in both either UEFI or legacy boot mode
So please run again a manual custom-installation, but do prepare nvme0n1 as such:
:->: ESP 256MB vfat maked as esp+boot
:->: "/"-root ext4 partition
:->: no boot partition.
+++EDIT: +++
And if you really want to install and boot only in legacy (BIOS) mode, do create a MSDOS-partition table,
which can hold the embedded legacy GRUB-loader.
Otherwise with GPT-partion table -as mentioned - you "must" (or better do so) to create a 1MB sized unformated partition marked as bios-grub/grub-bios, preferable as first partion on the drive.