Using domain.com, domain.net & domain.org

You would map those domains.

https://www.ads-software.com/extend/plugins/wordpress-mu-domain-mapping/

I have the mapping plugin installed but the site has to be created first and then mapped. That means I have to create a second site with the same name (domainexample.mainsite.com) and then map it to domainexample.org or domainexample.net. I didn’t think WordPress would let me create a second or third site with the same name domainexample.

That means I have to create a second site with the same name (domainexample.mainsite.com)

You don’t have to use the SAME name.

You can name it ‘domainexampleorg’ ??

I have the mapping plugin installed but the site has to be created first and then mapped.

Look at the settings page for the domain mapping plugin. you CAN map another domain to the SAME site.