Is the third-party site and the referenced PDF available on the open internet? If so, then please post the URL.
It would appear something has co-opted the normal Preview-based PDF view that's provided within Safari. (That PDF display process doesn't involve Java by default.)
There are stability and security updates for issues with various Adobe tools. Confirm that your Adobe Acrobat and other related Adobe pieces are patched to current.
Safari with the current Java patches defaults to disabling Java, and will also disable Java if it is not being used regularly. To enable Java within Safari, select the Safari menu > Preferences > Security > check the Enable Java box. (Whether this setting might effects the particular third-party web site depends on what the third-party site is doing; whether the site is using Java.)
Java downgrades? As was mentioned earlier, the easiest is to roll in your current backup. If you don't have a backup that was created prior to the installation and if the update does not work, then you have a problem as AFAIK there isn't an easy way to remove or downgrade Java. That may end up being a reinstallation of OS X; of rolling out a backup (or two) onto an external disk (or two), wiping the system disk, and reloading everything except the latest Java, including migrating in the contents of the external disk.
If you have AppleCare, you can call Apple and ask for help troubleshooting this.