From the picture, your EFI partition appears intact, and still shows as Fat32.
The Microsoft Reserved (MSR) partition is giving different indications than mine. It indicates on my system using GParted as Unknown and msftres. Did you change it at all? We probably need to get it repaired to show the correct information.
When you check the bios during boot, there is no Windows Boot Manager option?
The Shimx64.efi file is one used to make the install compatible with the Windows 8 Secure Boot. In the latest version of Ubuntu, it is no longer needed.
A Legacy version of Linux is able to use a partition it makes to boot using MBR on a GPT configured drive, so that is what you are seeing. During boot, if the drive itself is selected as the boot device, it will use the MBR and boot to Linux. If the Windows (or Linux) Boot Manger is used, it should boot to the EFI version of the OS.
I also see you have another Fat32 partition. If it is on the Hard Drive and not the Flash drive, it might be confusing Windows since it could look like another EFI partition. It may not be necessary, but can you copy the contents somewhere else and remove it?
Now, if you are not showing a Windows Boot Manager in the Bios, you might run the bcdboot c:\Windows command to replace the boot files. Not sure what might happen to the Other OSes during this process, so you may want to hold off until you decide.
Edit: I just noticed your EFI partition does not show as the System Partition, but the Recovery Partition does. Not sure what might have happened there, except you may be booting through the recovery Partition or even labeled it that.