Good dedicated server hoster?

The question is already up.

Purpose: Minecraft server with one player growth

What budget?

https://www.soyoustart.com/de/game-server/

Look here

Price / performance are the top have very good single core performance with the things which is very important for Minecraft

I've been with hetzner for 12 years now, only to be recommended.

Dedicated?

Would start small.

Good and cheap vServers are available from Contabo

For dedicated I would look at netcup or serverdiscount