OK. Let me try to help you in points:
1. Otto’s guide is TWO AND A HALF YEAR OLD. It is now nearly completely out-dated. Plugin has evolved by that time and has been updated many times. That is, why you see some many differences between those screenshots and messages there and real plugin. For me, instructions at installation tab of plugin itself:
https://www.ads-software.com/extend/plugins/wordpress-mu-domain-mapping/installation/
was enough for me. I was able to setup all the things and get it all working.
2. Image, you’re reffering to is Domain Mapping section, where you add actual domain, you want to map to your blog site. You can access it via Tools > Domain Mapping. But it is available there only for all your network-created sites. That is: only for those blogs that were created after you finished setting up your blog network. This option is not available for Network Admin and not for blog (site) you had installed as first one, before creating a network.
3. Maybe most important. Due to (probably) some bug in this plugin, section Tools > Domain Mapping is not avaiilable (visible), if you network activate this plugin — i.e. make it active for all your sites. Deactivate it (click on Network Deactivate in Network Admin > Plugins) and instead go to each of your sites (except: as above) and manually activate this plugin for each of your sites / blogs. Only then this option become available and you’ll be able to access screen, you’re showing on screen-shot.
4. If, after clicking activate (either network-wide or site-alone one), you see a message about fatal error about redeclaration of some functions, then this 99,99% because I have read (and did) whole OUTDATED Otto’s blog and copied a file into mu-plugins directory. This is DEPRECATED, you don’t have to do this anymore. Keep yourself with installation guide on plugin page and forget about all this mu-plugins stuff!
5. I used IP address instead of CNAME in plugin configuration and it seems to be working for me just fine. That is all for plugin. As for your domain (DNS) configuration: you probably should use both CNAME and IP address in domain config to be able to access both example.com and https://www.example.com. I wrote “problably”, because I haven’t tested this out. It works just fine for me, no matter if I use my domain with or without https://www., so I haven’t even checked, what do I have configured in my domains. If you don’t need https://www., then you can forget about all this CNAME stuff.
