So I have checked deeper into the 2D and 3D online maps and came to the following findings:

1. Minecraft does not update the worlds to 1.13 when the server starts but only when an area of the world is actually loaded.The rest of the map is still 1.12 until someone actually goes there.
   
2. Mapping tools, be it for 1.13 or 1.12 can only read their respective version. If I use an old mapping tool, it will show only areas that have not been visited since the upgrade, if I use a new tool, only areas people visisted since the upgrade are visible.
3. The tool for the 3D map (dynmap) has been properly upgraded. I am right now re-rendering the worlds. The city for example is done, but you can see that there are empty areas. Unless we find or create a tool that will load & update the blank areas to 1.13, someone will have to visist those areas to get them updated and therefore enable them for rendering. This is not so much an issue in the city, but larger worlds like the kingdom are much more affected.
4. The tool for the 2D map is working only half-way. I have contacted the author with a list of bugs I found and I hope he will be able to help fixing those. Otherwise I will have to do some manual workarounds to get things done. It will take some time and this is an issue since new users will have troubles finding lots with the current incomplete maps.