In the B-R log, we see that the BIOS is correctly setup to boot the HDD in UEFI mode:
We also see:
=================== UEFI/Legacy mode :
BIOS is EFI-compatible, and is setup in EFI-mode for this installed-session.
The presence of /EFI/Boot/bootx64.efi.grb means that Boot-Repair copied /EFI/ubuntu/grubx64.efi to /EFI/Boot/bootx64.efi (in case the BIOS is hard-coded to boot into /EFI/Boot/bootx64.efi).
Same for bootmgfw.efi and bootx64.efi.
The fact that partitioning is MsDos (not GPT) should not be a problem, as the UEFI specs don't require GPT (@Fred: i sent you a PDF with recent UEFI official specifications). And your case shows us that it is possible to use UEFI on MsDos.
However, UEFI on MsDos is not common (it is the first time I see it), and may not be reliable. This would explain why it sometimes fails to boot in your case, but the problem may be something else.
If i were you, i would format the disk GPT via Gparted, then reinstall Ubuntu, and see if this solves the problem.