Try to use the live system again and then use the “chroot” command to change your root to your actual install, mount the efi partition, and then try the grub script.
If all else fails try without grub, for example debian kernels can be copied to the esp directly and started from there, maybe ubuntus can too. check for example if systemd-boot can be installed on your ubuntu instead of grub
rEFInd can only boot EFI executables, and only on filesystems it can read. The linux kernel can masquerade as an EFI executable, and when dumped on the ESP be booted. But you are probably still better off using something like systemd-boot unless you know how ubuntu updates work on a technical level to make it copy the kernel image each update 
Thank you very much for your reply, which I greatly appreciate. Although I understood your explanations, I still needed to do further research to read up on the topics in more depth, which I did. However, this is a level that I am not really familiar with and requires very detailed knowledge. Since the attempts according to the instructions did not work right away, I gave up. It is very complex and I would probably need weeks to really understand it all.
And since the Ubuntu (Unity) installation is not that old, I took the risk and simply reinstalled it. And what a surprise, rEFInd, Haiku, and now Grub/Ubuntu are running again. I didn’t have to do anything to rEFInd, it just worked, and so did Haiku. I tried several reboots in both systems and it always worked. I hope it stays that way. I made a copy of the EFI folder of the EFI partition just to be on the “safe” side in case this happens again. Thanks anyway for the efforts I got here.