First one must be aware that the given translation of the hadith includes an interpretation (a fiqh view which was concluded from the original text) as the word Mazameer مزامير refering to instruments that are like a flute or oboe was interpreted as "music instruments". SO one may say it is a generalization which is not literally given in the original text!
This hadith has been quoted in different hadith compilations. Among them is al-Mustadrak of al-Hakim from Nishapur.
Note that the link of al-maktabh here adds the comment that imam a-Dhahabi didn't quote it in his at-Talkhees (a summary of al-Mustadrak). This might lead to the conclusion that he didn't consider it sahih. But as it is usual among scholars of this time and later we may strongly assume that a-Dhahabi compiled this book at the beginning of his career especially as he changed his mind about many qualifications he gave there. Also note that al-Hakim didn't add any information why he compiled this narrative. He usually would add a comment on whether a hadith fits into the conditions of al-Bukhari or Muslim or both etc. but he didn't say anything here after quoting the hadith.
The narrator chain of this version actually is one that would lead to the conclusion that this hadith might be sahih as all narrators are trustworthy, but ibn Abi Laila is considered as a person who had a bad memory nevertheless scholars consider such a narration as a possible evidence so it can be considered as at least hassan due to the amount of confirming narrations and narrator chains. According for example al-Albani in his silsilat al-AHadith as-Sahihah wa Shay'un min fiqhiha wa Fawaidiha سلسلة الأحاديث الصحيحة وشيء من فقهها وفوائدها says see here hadith #427 page 790 where he added several other narrations and versions of the hadith which are less reliable:
For example in the version of Ar-Ruba'iyat of abu Bakr a-Shafi'i has a fabricator in the narrator chain. Al-Mundhiri and al-Haythami considered the version of a-Diya' al-Maqdisi as hassan as it deviates and has two roots to a-Dahhak. Al-Bazzar and ibn Sa'ad actually compilled it with a similar narrator chain as is presented in al-Mustadrak.
Imam a-Thirmidi compiled a different version with a deviation in the content (see here) that would need much interpretation in order to make it refer to music, but has the same root.
At least the three linked versions in your post (al-Mustadrak, As-Sunan al-Kubra, Musannaf ibn abi Shaybah) all have the same root to ibn Abi Laila,
?...? -> ibn Abi Laila -> 'Ata'-> Jabir ibn 'Abdillah -> 'Abdurrahman ibn 'Awf
but differ from there on so some of them may have less reliable narrators than those of al-Mustadrak.
Finally al-Albani considers this ahadith as another evidence against ibn Hazm's view of declaring music instruments halal as Mizmar (a flute or similar instrument) is such an instrument.