Reinstalling pre-installed macOS X on MacBook Pro

MacOS NEVER allows installing Older over newer.

The only way you can install older is if you completely remove the newer. On a more recent Mac, you could depend on the built-in recovery partition in your mac's ROM. that is NOT Present on a 2010 model.

Time machine backup drives after 10.7.3 are said to contain a recovery partition as well, but you need to CHECK that before you bet everything on its working.

if you have a Bootable USB-stick installer of Mac OS 10.6.7, TRY IT before committing. It should boot up, offer to install, then eventually tell you no drives qualify because your internal drive has a NEWER MacOS on the internal drive.

at that point, presuming you trust you backup, back out if the installer (or start again) and invoke Utilities from its menu BEFORE you commit to installing. Use Disk Utility and Erase the drive. (you may have to partition as one partition as well) Then Install from the USB-stick, and you end up running a MacOS OLDER than 10.6.8, and can either restore everything INCLUDING macOS, or just restore your stuff, and manually software update back up to 10.6.8.

if you have an external drive, you could install on that external drive instead if you would feel safer.

can you show a Photo of the Disk Utility window with the internal drive selected?

if you have an internal SSD, 10.13 wold have reformatted the internal drive to APFS format, which MacOS 10.6 does Not understand. (This could be related to the Time machine issues as well.)

You would need to have selected the drive by the immutable manufacturer-given Device-name, and Erased THAT, specifying NOT 'as currently formatted' but instead GUID partition map and MacOS Extended (journaled) file system.

It is possible it is demanding an Apple drive inside that Mac.

if that were the case, you could install on an external drive, hen use software like CarbonCopy Cloner to duplicate the image onto the boot drive.

MacOS Base System is a DISK IMAGE. In this case, it is the Mounted minimal MacOS used to run Recovery and Installer. it is showing total size of 2.01GB. that is far to small to even THINK of installing onto -- you need at least 64GB and preferably WAY more.

That is the 'drive' you are running Installer and Disk Utility from. It's NOT in your way.

I do NOT recommend leaving your drive named Untitled. The default from the factory is Macintosh HD if you can't think of a better name, use that.

if this were failing during install, you could look in the Installer log (it's there, you just have to look around for it)

but since it's refusing to install, I don't know if there is a log showing WHY.

yes. that Utility is Disk Utility.

post a photograph of its screen

That looks right.

Is that the way it was when it refused to install?

if yes. I suggest you Restart and try install again.