Internet recovery mode. Difference between disk0s1 and disk1s1?

So then what you do first is open Disk Utility and select the top most partition after using View menu to show all of them.

Then hit erase.

Select APFS if available, if not select HFS Extended GUID.

Quit Disk Utility and run the installer.

Use internet recovery only if you plan to wipe the system or create a new partition with the original OS that came with your Mac, or replace the original storage itself after failure.

Use the partition option if you want to keep whatever is presently installed, and have space to put on a second system.

The disk#s# refer to various partitions. A partition is a uniquely definded group of sectors on the hard drive that can be used to define multiple unique formatting for the purpose of dual-booting two operating systems, or keeping a section of the hard drive available for erasing after archiving data away to two other external sources. On some Macs, such as those with a Fusion drive, the partition also separates the solid state from the platter part of the hard drive. Generally those are only found on iMacs, and generally the partition scheme is more smooth. There are directions on Apple Support to merge split Fusion drives, if you run into any.

Well not exactly. It gives you a fresh copy of the original OS. Then you need to run the installers up to the max OS after determining what version that is using http://checkcoverage.apple.com to find the age of your computer and its model.

http://www.everymac.com/ will tell you the maximum OS.

command-R does the currently installed OS, if you erase the partition itself from recovery, and then run the installer.

I believe that is controlled by the View menu selection. Regardless, are you trying to recover the original OS that came with the computer? Do you have any data you wish to preserve?