"Macintosh HD" boot partition gone

You can name the drive whatever you want but the usual name for a boot drive is "Macintosh HD." You are only dealing with the topmost item labeled, "...Apple SSD..." We call that the out-dented entry. It is the physical device. The entry below it we call the indented entry. It's the logical volume, usually name "Macintosh HD." When you reformat or erase the drive, the only entry that is changed is the indented entry or the logical volume. Once you provide a name, it will not change with each reboot.

After you erase the drive you will have an indented entry now named "Untitled." You can change that name before continuing with the installation of macOS. I suggest you provide a name such as "Macintosh HD." When you reinstall macOS, choose that volume as the one on which macOS will be installed. Your configuration in Disk Utility would then look like this:

121.33 GB Apple SSD T...

Macintosh HD

The entry you now see, disk0s2, will be gone.

1. Restart the computer. Immediately, at or before the chime, hold down the (Command-Option-R) keys until a globe appears.
2. The Utility Menu will appear in from 5-20 minutes. Be patient.
3. Select Disk Utility and click on the Continue button.
4. When Disk Utility loads select the target drive (121.33 GB Apple SSD T...) from the side list.
5. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
6. Set the Format type to Mac OS Extended, (Journaled.)
7. Enter a name for the volume. I suggest using "Macintosh HD."
8. Click on the Erase button and wait for it to complete.
9. Quit Disk Utility and return to the Utility Menu.
10. Select Install OS X and click on the Continue button.

If you do a network recovery, it works like so:

Internet/Network Recovery of El Capitan or Later on a Clean Disk

     If possible, back up your files before proceeding.

1. Restart the computer. Immediately, at or before the chime, hold down the (Command-Option-R) keys until a globe appears.
2. The Utility Menu will appear in from 5-20 minutes. Be patient.
3. Select Disk Utility and click on the Continue button.
4. When Disk Utility loads select the target drive (out-dented entry w/type and size) from the side list.
5. Click on the Erase button in Disk Utility's toolbar. A panel will drop down.
6. Set the partition scheme to GUID.
7. Set the Format type to APFS (SSDs, only if installing Mojave or Catalina ) or Mac OS Extended, (Journaled.)
8. Click on the Apply button, then click on the Done button when it activates.
9. Quit Disk Utility and return to the Utility Menu.
10. Select Install OS X and click on the Continue button.

This should install that version of macOS that had been installed. However, because you erased the entire drive, you may have destroyed the Recovery HD. So, you may find Mountain Lion installed. You can then upgrade to High Sierra if your model supports it. You did not provide your machine information.

Yes, you have made the correct selection. You are using an older version of Disk Utility. You want to make the setting as show in your second image, Mac OS Extended, Journaled. You will simply omit Step 6. In Step 8 you will click on the Erase button.

I'm afraid I did not quite understand, "...this went back into time so many years after restarting and now I am stuck with this old menu..." You're doing just fine, as far as I can tell.

Everything I see in this image is as it should be. There are two icons on the left of Container disk2. The first one is the triangle which is used to "open" the container's contents. The second icon is just a disk icon or "volume" icon. The different disk icons from those under Disk Images is because they are ejectable media. That is why they have the Eject icon on the right of the Apple disk image Media entry.

You cannot eject an internal disk, thus, there is no Eject icon to the right of the Container or its sub-volumes. With APFS and Catalina there are now two volumes shown in Disk Utility - Macintosh HD and Macintosh HD - Data.

The first volume is locked and only contains the OS. It is locked to prevent any changes. This provides added security against malware, and it also prevents user changes that could also damage the OS. The Data volume contains everything else including user accounts and all user files.

On the Desktop the only disk you will see is Macintosh HD, which is actually an alias that displays what you would normally expect to see. The Data volume is invisible. There are three other invisible volumes in the container that are not displayed by Disk Utility. These are Preboot, Recovery, and VM. The Recovery volume is what used to be called the Recovery HD. The other two are reserved for use by the OS.

All of this represents the basic structure behind APFS formatting under Catalina. I hope this explanation is helpful.

No, it doesn't work quite that way. If you desire to do a true Erase and Install, then you need to remove both volumes so as not to confuse the installer. Now, if you just wish to erase the Data volume then restore its contents from a backup, then you can do that without removing either volume. This assumes no change in the version of macOS as the basis for a clean install.

The new structure essentially protects the OS from damage by anything other than a disk corruption problem. However, since the system volume is locked, it's unlikely that the disk can be damaged by a bad write operation.

I use Carbon Copy Cloner for my regular backups. It can backup and restore both volumes in a single operation. It also can backup and restore just a Data volume. I'm not familiar with how Time Machine handles Catalina backups other than they are different from previous OS versions, and they cannot be intermixed with backups of older versions. Fortunately, CCC handles the new structures perfectly.