What is the difference between Minecraft Freebuild and Citybuild servers, one builds somehow in the open world (what I'm looking for) and the other builds on PlotMe worlds as far as I know. But what is what and are there other terms that describe these server types, e.g. If the server still offers a / claim option?
So FreeBuild runs all over one world. You farm and build your base in one world. The dangerous thing about this game fashion is that you can be discovered and grooved at any time. CityBuild is the more regulated version of Freebuild. There's a farm world in which you farm and a plot world in which you build your house. Everything is secured.
Are there servers where there's a farm world and then an open world where you can build?
Yes there are such servers. You can then build a house anywhere in the world and secure it. Unfortunately I don't know the name of such servers. I only know the new server that is offering these days to attract new players. Just have a look at minecraft-Server.eu. There you can look at new servers
But then you can't search for a specific type of server or, since we don't know what its name is exactly?
The difference has been well described. But there's still the option to have a plot of land fully protected and to be able to mine there without a farm world. This is also called freebuild because you can choose the place yourself, put down a box and have a predefined size that can also be enlarged with a gold scoop.
There are many variants of freebuild, it does not always have to be something unprotected or plot land.
It is best to always read the description of the server, since it often shows what options you have.
Citybuild Nature Griefergames xD
Go to CB Nature from Griefergames, you have such an open world where you can get a small part of a world to build. Everything secured. And then there's a farm world xD