Mac Os Monterey not showing in update and app store

please you may use this page's instruction to download Monterey installer file.

i suggest to get the 12.6.1 , as thats more stable, (as of now Jan,2023), than the 12.6.2

so change the linked page's word "10.15.7" into "12.6.1"

when the installer downloads, just run the installer app file.

Although a Terminal command to manually download Monterey was provided, it just will not wok - period.

The computer, as presented in your earlier posting denotes this computer is a MacBook Pro (Retina, 15-inch, Late 2013)

It suggested these review the list of Qualifying Computer from the below link

refer to link below

macOS Monterey is compatible with these computers - Apple Support (CA)

Excerpt from above link

• MacBook Pro (Retina, 15-inch, Mid 2015)
• MacBook Pro (Retina, 13-inch, Early 2015)

If the user whats the latest and greatest versions of macOS, all is not lost.

If the user wishes to explore the Apple Trade-In  Process where the Value of the exiting device could be used against the cost of a new Apple Computer, iPhone or iPad  

Or Shop Refurbished Apple computers at an Apple Reduced Cost

Have used both previously and have been satisfied each time.

the "$" symbol itself is not a command ... its command-prompt symbol for "Terminal" app,

... like windows-command-prompt had the ">" symbol & it displayed/displays C:\ >

... similar in left of the "$" symbol, in Terminal app you will see your folder/directory location has the "~" (tilda) symbol ( instead of C:\ ).

... in macOS (macOS uses DARWIN, BSD, etc UNIX OS componenets) , the "~" symbol is an abbreviation, it means your "HOME" directory , which is the located here : \Users\<your-user-name-in-macOS>\

Big Sur is the latest version of macOS you can run on a Late 2013 MBP.