If you do a fresh install, don't install Java manually as we did, I'd like you to test something.
EDIT: to be more specific, the tutorial you gave refered to another one used to install minecraft on another microcomputer from nVidia. While it seems to have an ARM CPU, it uses the PPA from WebUpd8 to install Oracle's Java, even though this one is supposed to have builds only for x86_64 CPUs.
So, basically, after you'vre installed Ubuntu MATE, try the following:
sudo add-apt-repository ppa:webupd8team/java
sudo apt-get update
sudo apt-get install oracle-java8-installer
and tell me if it works or if it tells you that it can't find the package.
I doubt your previous issue come from Java, though, as it seemed to be installed properly and Minecraft should work on the Pi with OpenJDK anyway (with very bad perfs but it should launch).