Checkmk itself is the same in both setups, relying on the MTA in the underlying system, be it the appliance or a “normal” Linux system.
The appliance is meant for people without Linux knowledge. It’s like a “Checkmk-specific Linux distribution”, ready to use. Looks like tribe29 did a good job there and have offered an easy interface for the MTA configuration. Of course it helps that nullmailer is a rather simple MTA that doesn’t have too many features anyway.
If you have a Linux server running a general-purpose Linux distribution that you install and administrate yourself, then you need to configure your MTA yourself as well. With the tools/methods your distribution offers. You could also use nullmailer instead of postfix, but that’s your own decision.
What your screenshot of the Global settings shows is the fallback address – that’s a destination address, not a from. Totally unrelated.