MacOS – Is it possible to choose which macOS version to use when reinstalling

The Mac OS X base system image is not the cause of your problem. See What is the 'Mac OS X Base System' disk image on my 2011 MacBook AIr? for an explanation.

My best guess, based on the screenshot, as to why you can't repartition the Hitachi disk is that it's the current startup device. You need to restart your computer from an an external device—preferably a device that has the Lion Installer on it-if you want to reformat the drive. Otherwise, assuming you have a valid recovery partition, you can erase the Mac partition itself and reinstall Lion without an external device.

Perhaps Apple's Mac OS X 10.7 Help: Recover your entire system document will help you.

In my experience a locked (read-only) disk in OS X means one of two things: An incompatible filesystem (Like Windows NTFS) or a dying drive. Yours is certainly not the former and likely the latter.

First, see if it can be repaired because it's worth a shot. Open Disk Utility from within the Recovery partition that it boots you into. Highlight your drive on the left-side column (You should see two hard drive icons. The top one represents the physical device and the lower one, indented, represents the volume that the computer sees) and then click the "Repair Disk" button on the right-side panel. See what it spits out at you in the log. Basically red text bad, green text good.

If it tells you the disk was repaired (green text): Restart the Mac and see if it boots up correctly. If it doesn't try holding option on boot and selecting your hard drive as the boot device. If you get back into the system successfully, BACK UP YOUR FILES with Time Machine or your preferred method to another disk.

If it tells you the disk couldn't be repaired (red text): Your drive might be failing. If you plug an external drive into the computer, you can use Disk Utility's Restore tab to clone your hard drive to the external drive. Source = your drive and Destination = the external drive (You can drag and drop them from the left column. Note: Make sure the external drive is blank because restoring wipes the destination drive before copying.

Once you've restored you could try booting off the external drive (Option on startup, select external drive) and doing the restore process in reverse.