I have the same issue. I support a large organization and we need a to know how to (1) access the jre 6u51 (both x86 and x64 for Windows) and (2) what our rights are for distribution.
In short, you can use Java 6U51+ only with the Oracle application/product that entitled you to it in the first place. This means that for example, if you are using Forms/Reports Service 10.1.2.3, all end-users required to access Forms from this installation can use Java 6U51. If the end-user is not using an Oracle product that requires this Java version then they are not entitled to have it installed on their machine.
See Note 1557737.1 for details.