MacOS – What physical storage does Internet Recovery use if the internal drive is broken

I am assuming that you got a flashing folder with a question mark similar to this:

That means your Mac cannot find the boot loader.

Now, since you said that you were able to go into disk utility, you already know how to boot into Recovery Mode. What I am assuming you saw when you were working in Disk Utility is something similar to this:

The SuperDrive being greyed out is probably because the drive is empty. disk0 is typically the disk drive in your Mac. Mac OS X Base System is more than likely a partition on disk0

Do this:

Hold down the Option while powering on your Mac. If there is a bootable partition, it will be listed (other than the recovery partition). If so, boot into it.

Once you have booted, check to see what you have selected as your startup disk (Preferences --> Startup Disk)

Thanks @Tetsujin. The listing looked a little screwy so I was a little hesitant to go mucking around with the command-line diskutil. But I ended up doing a

diskutil corestorage list

diskutil corestorage delete [UUID of 50GB Logical Volume Group]

GUI Disk Utility correctly saw the 50 GB of empty space as part of the other Logical Volume Group and I was able to delete it and merge it back in fine.