"I tried to do the same thing by installing the multisite in the "home/public_html" folder, "
Yes, for Domain Mapping to work, WordPress must be installed in the root of your account (e.g. home/public_html/) or the root of a SubDomain (e.g. created in cPanel). Installing WordPress in a folder won't work for domain mapping. Said another way, DNS has to take the URL directly to where WordPress is installed.
"- site-A (the main) ==> domain-A"
Not clear if you are trying to map the domain WordPress runs on. If so, Do Not. WordPress itself needs a "stable domain name to work on". If you need to undo it, I cover how in my Domain Mapping Instructions. (You'd know; nothing works.)
"Site ID: 2" -- Forgive me if I'm saying something you understand, but so many people don't...
From /wp-admin/network/sites.php, if you hover over the Domain you want to map, and it shows this for the URL "http://YOURDOMAIN.com/wp-admin/network/site-info.php?id=6", then you should put 6 in this field.
On /wp-admin/network/sites.php, do you have a column titled "Mapping" and does the mapping you defined show?