Reinstall OS X Mountain Lion 10.7

You need to partition & format the physical drive which seems to be "Media" in the screenshot. This is very unusual, but I also have not used macOS 10.7 in many years. Here are instructions for partitioning & formatting the physical drive for installing macOS 10.6 to 10.10:

https://eshop.macsales.com/tech_center/formatting/Mac_Formatting_6-10_R3.pdf

After partitioning & formatting the drive, quit Disk Utility and select the "Install macOS" option. If you reboot the Mac without selecting the "Install macOS" (or possibly "Reinstall macOS" option), then you won't be able to boot into Recovery Mode again unless you are currently booted into Internet Recovery Mode.

You may need to configure the system date to a much older date as the certificate of the installer may be outdated.

If macOS 10.12.6+ was installed at some point in the past, then theoretically it should be possible to access the online macOS 10.13 installer using Command + Option + R, but unfortunately sometimes it will still only boot to oldest macOS online installer compatible with the Mac.

If you have access to another Mac from Late 2009 to 2018, then you can use that Mac to create a bootable macOS 10.13 USB installer using the instructions in this Apple article:

Create a bootable installer for macOS - Apple Support

The only important screenshot is the first one with Disk Utility as it shows what appears to be an internal drive called "Media" which has no partitions or volumes on it. If that is the physical internal drive and it has no space, then that drive has most likely failed and needs to be replaced. If the item named "Media" is not a physical internal drive, then it that means the internal drive has failed or is missing.

"disk1" which shows a volume with orange icon "Mac OS X Base System" is just a virtual volume associated with the installer. The orange icon indicates "external" media.

To attempt to get more information about the drives, launch the Terminal app from the Utilities menu and execute the following command:

diskutil  list

If the output scrolls off the top of the screen, then use this command instead so it does not automatically scroll past the screen:

diskutil  list  |  less

Take a picture of the screen. If the screen is full, you can use the down arrow key to scroll the output to display more of the output....take another picture, until everything is photographed. Usually the physical media should be the first items displayed, but if the drive is failing, then the drive may show up later in the output. Usually the first item will be the internal physical drive identified as "disk0"...any entries before "disk1" are probably the most important & relevant here. Unfortunately macOS Lion does not have any commands to filter the output to just display the internal drives.

What happened to get you here in the first place? It may be relevant since the same issue may be why you cannot do anything to the internal drive...assuming the item seen as "Media" is actually a physical drive.

tmaprez wrote:

There is no Terminal app in the Utilities menu, so I cannot launch that command.

While booted into Recovery Mode click on the "Utilities" menu on the menubar to access the Terminal app. AFAIK, this has been available for every version of macOS, but it has been a while since I have used OSX 10.6 or macOS 10.7.

The system is so old I wanted a fresh install since I've already transferred everything over to my MacBook Pro. If nothing else, I wanted the iMac for light use only, I.e. Safari, Acrobat, etc. So after I was certain I had transferred & backed up everything I reformatted the HD from Disk Utilities. Did I erase the wrong drive?

No idea. It is possible the hard drive is failing. Since no volumes are shown on the internal drive in your pictures of Disk Utility, you will need to use Disk Utility to partition & format the internal drive. Use the instructions in this article on how to partition & format the drive for use with macOS 10.6 to 10.10:

https://eshop.macsales.com/tech_center/formatting/Mac_Formatting_6-10_R3.pdf

Any other suggestions would be appreciated!

Generally if you have access to another Mac from 2008 to 2018, then you can use that Mac to create a bootable macOS 10.11 or 10.13 USB installer which will probably be the easiest & simplest way of reinstalling macOS to this 2010 Mac assuming using Command + Option + R to access Internet Recovery Mode does not boot to the online macOS 10.13 installer.

Otherwise you will need the original OSX 10.6.3 Snow Leopard Restore DVD which originally shipped with your 2010 iMac from the factory. Once OSX 10.6.3 is installed, make sure to update the OS with all its patches. Then you should be able to download and install macOS 10.13 (or at least macOS 10.11 since some people report they may need 10.11 before being able to go to later versions of macOS if 10.11 was never previously installed on the system).

You can try installing macOS to an external drive, but with a 2010 iMac it will boot very slowly due to just having USB2 ports. Replacing the internal drive is a bit difficult as there are several very delicate & fragile cables/connectors which are both hard to see & reach which are difficult to reconnect. Check out OWC for internal storage upgrade options (an internal SSD would improve performance), but you will need an adapter bracket and a thermal sensor adapter as well which OWC provides in their iMac SSD upgrade kits. OWC also provides video instructions for the process which you can view prior to purchase of the upgrade kit, but keep in mind the video does make things look easier than they really are.

FYI, I would recommend trying to boot into Internet Recovery Mode using Command + Option + R (no shift) to see if you can access the online macOS 10.13 installer which would make things easier.

You also have to consider whether the upgrade is worth the cost for such an old iMac which is restricted to macOS 10.13 as the highest OS compatible with the 2010 iMac. While third party browsers are still supported on macOS 10.13, it probably won't be too long before they will no longer be supported. Safari for macOS 10.13 is already too old to work with most of today's websites. It is possible to install Linux Mint on to this iMac which would give you access to the current versions of many popular web browsers (Firefox, Google Chrome, Vivaldi). Using Linux is a great way of extending the useful life of older computers, but it does require you to learn a new OS. The popular third party apps such as Microsoft Office, Adobe, etc. are not supported on Linux by their developers although there are usually other alternatives which may suffice such as LibreOffice, etc. I believe Linux Mint has an option on their installer to run Linux Mint from the USB stick to give you a basic idea what it is like, but performance will be extremely slow from the USB stick.

It appears that your internal drive has failed since it does not appear on the list.