Changing the value of the box that says Level Name on the panel isn't for the name of the world. That is just the name of the world the server takes you to when you join the server. So if you changed that, it will just generate a new world and put you in it.
If the world is just the default world, you can do /dl world to download it but this won't work for the nether and end, if you want to download those too.
If you want to download the nether and end too, you can use a plugin called Multiverse to clone the world to have a different name. With the plugin, first do /mv clone "world name in quotes" newName. This should clone that world with the new name. If it says Multiverse cannot recognize the world, then do /mv import "world name in quotes" NORMAL. Or if its a nether world, do NETHER, and END for end and then run the clone command. After that, just download the cloned world.