The Frankfurt edition of Sefer Hachinuch, published in תקמ"ג/1783, introduced many errors into the text, involving changing the wording, the ordering, and adding whole new sections. You can see Rabbi Chayim Chavel's summary of changes they made here (and examples of specific changes a few pages later). These changes were copied in all of the editions that followed it.
I believe the "uncensored" edition is probably a reference to the editions without the changes of the Frankfurt edition.
I think that Rabbi Yaakov Kaminetsky was most likely reading the Mosad Harav Kook edition of Sefer Hachinuch, edited by Rabbi Chavel, and first published in 1960. This edition was based on the first edition and included notes about the changes of later editions. Alternatively, he might have somehow found a facsimile of the first edition.