It sounds like the cable may not be folded properly, as it should fit just fine into the left socket. If that isn't able to fix the issue, you may want to contact the company you got that cable from.
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== Update ==
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Thanks. The problem starts with the original hard drive flex cable (not the 2nd drive kit cable). The original cable that I saw so far on the internet connects to left connector, however, when I open my mac (brand new not refurbished), it is connected to the right connector. This causes conflict because the 2nd drive kit cable (ifixit and Oxx) also connects to the right connector. My question is whether other experience the same thing and Apple actually made a design revision on the 2012 mac mini.
It sounds like the cable may not be folded properly, as it should fit just fine into the left socket. If that isn't able to fix the issue, you may want to contact the company you got that cable from.