You want jre. Just do a search for jre and it should give you several
choices from open jdk to sun.

Yes, you are right. Synaptic is more complicated, but that is good. You
can't have too much information if you want to learn. Just ask questions if
you don't understand. The advantage of Software Centre is that it
simplifies, but that is at the expense of knowledge and dumbing down users
eventually works against them.