Mac doesn't boot after reinstall from back up

OK. That is all helpful. The first problem means either Catalina is already installed on the drive or the installer is on the drive. The second problem means the SSD is not formatted correctly. So, you need to do the following:

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

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, type and size info) 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).
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.

If you have another Mac available, you can make a bootable installer flash drive to install Catalina. Download the free utility DiskMaker X 9.0 that you can use to create the bootable flash drive. You will need a 16GB USB flash drive formatted Mac OS Extended (Journaled).

If you are OK using Terminal, then you can follow these instructions. For either process you will need to have the macOS Catalina Installer located in the Applications folder.

Bootable USB Installers for OS X Mavericks, Yosemite, El Capitan, and Sierra

First, review this introductory article: Create a bootable installer for macOS. Second, read this How To outline for creating a bootable macOS installer. Simply use the Terminal command by copying and pasting the command line for the version of macOS from the list below. You will need a 16GB or larger USB flash drive that has been partitioned and formatted for use with OS X as well as an installer for the desired macOS version that you have previously downloaded from the App Store. The installer must be located in the Applications folder.

Flash Drive Partition and Format

1. Open Disk Utility in the Utilities' folder.
2. After Disk Utility loads select the drive (out-dented entry with the mfg.'s ID and size) from the side list.
3. Click on the Erase tab in the Disk Utility toolbar. Name the drive, "MyVolume". <---- IMPORTANT!
4. In the drop down panel set the partition scheme to GUID. Set the Format type to Mac OS Extended (Journaled.)
5. Click on the Apply button and wait for the Done button to activate. When it does click on it.
6. Quit Disk Utility.

Create Installer

Open the Terminal in the Utilities' folder. Copy the command line (begins with "sudo") and paste that entire command line at the Terminal's prompt:

Command for macOS Catalina:

sudo /Applications/Install\ macOS\ Catalina.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume

Press RETURN. You will be asked for your admin password. It will not echo to the Terminal window. Then press RETURN again. Wait for the return of the Terminal prompt signifying the process has completed. It takes quite some time to finish. Be patient.

The bootable installer can be used to install Catalina. Shut the computer down. Insert your flash drive into a USB port. Start the computer like so:

Boot Using OPTION key

1. Restart the computer.
2. Immediately after the chime press and hold down the  "OPTION" key.
3. Release the key when the Boot Manager screen appears.
4. Select the disk icon for the USB flash drive.
5. Click on the arrow button under the disk icon.

How to choose between APFS and Mac OS Extended when formatting a disk for Mac. Have you actually tried a non-erase reinstall from the Recovery HD?

Reinstall El Capitan or Later Without Erasing Drive

Please be sure you back up, if possible.

1. Restart the computer. Immediately, at or before the chime, hold down the Command and R keys until the Apple logo and progress bar appear. Wait until the Utility Menu appears.
2. Select Disk Utility and click on the Continue button.
3. Select the indented (usually, Macintosh HD) volume entry from the side list.
4. Click on the First Aid button in Disk Utility's toolbar. Wait until the Done button activates, then click on it.
5. Quit Disk Utility and return to the Utility Menu.
6. Select Install OS X and click on the Continue button.