Is it permissible to rule a hadith in Al-Hakim's collection as authentic based on the authenticity of a similar hadith in Sunan Abi Dawud? And is there a difference, according to Al-Albani, between the terms "its isnad is authentic" and "authentic hadith"?

It is permissible to judge a Hadith in Al-Mustadrak as authentic under certain conditions: that the Companion who narrated the Hadith in Al-Mustadrak is the same Companion who narrated the Hadith in Abu Dawud, that the chain of narration (isnad) in Al-Mustadrak does not contain any accused, abandoned, or fabricator narrators, and that the Hadith in Al-Mustadrak does not contain any ruling or meaning that exceeds what is found in the Hadith in Abu Dawud, in addition to the condition that the Hadith in Abu Dawud itself is authentic.

The difference between saying "its isnad is authentic" and "the hadith is authentic" is that the latter is more specific than the former and entails it. Judging a hadith as authentic applies to both its isnad and its text (matn), whereas judging the isnad as authentic does not include the text. An isnad may be authentic, but the text may not be due to a hidden defect ('illah khafiyah) or anomalousness (shudhudh).