Is Java back in Mountain Lion?

No, Mountain Lion does not come with Java unless you install the Java update. But JavaScript is included in Safari.

I was under the impression that Java no longer came on Macs.

Apple no longer distributes it. You have to download and install Java on your own.

The Java Visual VM is present in ML but you must install a Java runtime to use it:

To add, if you run programs that require Java, and you click the Install button on that dialogue that John Galt mentioned, then you will download and install Java SE 6 from Apple. This is not the latest version of Java, which Oracle develops, but is one that will work with many applications that have come to rely on Apple's implementation of Java to work. Oracle's implementation does not use the same installation so installing it alone may still require you to get Apple's Java SE 6 runtime.

I do believe that applet is for Javascript, not Java.

Evil Ash wrote:

So all the stuff I saw in Safari's help<installed plugins that said java applet stuff was not a working version of java?

If it appears in Installed Plug-ins then it is installed. If you don't want it get rid of it.

I would attach a screenshot but no Java appears in any of my Safari's Installed Plug-ins and JavaScript is enabled. Despite their names the two are not related.

Java is probably installed at

/Library/Internet Plug-Ins/

or

~/Library/Internet Plug-Ins/

To locate each one select (triple-click to select the entire line) either of the paths above, then control-click and select Services > Reveal. Open the folder and dispose of the suspected plug-ins in accordance with the following: Safari: Unsupported third-party add-ons may cause Safari to unexpectedly quit or have performance issues

All I can suggest is that a website required Java, and you or someone else installed it.

Make Safari's Preferences look like this and Java will be rendered inert (in Safari, that is):

I haven't used Java in years and haven't found a way to install it inadvertently, so I have no other explanation for its appearance.

Merely restoring OS X doesn't remove components already installed, so if Java had been installed in the past it would remain installed. That would be the simplest explanation.