Hard Drive doesn't show in Recovery Mode

Solution 1:

'Going back' & 'update' are mutually exclusive terms.

You will need to re-format the drive to HFS+ first from Disk Utility. El Capitan cannot read or understand the APFS disk format that Catalina uses.

Your edit suggests you are still in Catalina & that your disk is currently APFS.
You will need to start from your external El Capitan boot medium [or from Internet Recovery] in order to be able to format your boot drive [you cannot format the drive you are booted from, for obvious reasons.]
Select the Physical disk in the left column [Apple SSD] & select Erase. Set to GUID/HFS Extended, Journalled.