DOC 9432 Manual of RT (4th ed 2007) is just like CAP 413
7.3.1 examples
"Cleared straight-in ILS approach runway 24 report established"
"Fastair 345 established, runway in sight"
"G-AB report established on the localizer"
"G-AB established localizer"
7.5.1 radar vectored approach uses the same phraseology as in the example above. So it's not just the UK that has the ambiguity.
I don't think PANS-ATM 8.9.4.1 is giving a phraseology example, just guidance as to the reporting criterion. All in all, I'd interpret "report established" as meaning "report established on the localizer", but I agree that there's ambiguity.