Okay so my reasoning as to why minato isnt a jinchuuriki was off, however it still doesnt discredit something important. Minato is actually a pseudo jinchuuriki instead, & to go even further than that, naruto is a pseudo jinchuuriki as well, neither of them are actually jinchuuriki who by definition are supposed to house a tailed beast, not part of it, let me explain:
The Yin Release (陰遁, Inton; Viz "Dark Style" or "Shadow Style") techniques, based on the spiritual energy that governs the imagination, can be used to create form out of nothingness.
The Yang Release (陽遁, Yōton; Viz "Light Style") techniques, based on the physical energy that governs vitality, can be used to breathe life into form.
When looking at the definitions, my initial thought before "yin" kurama was "Well if minato only possessed the spiritual energy that is also present within the uchiha, then it would only make sense that yin kurama would have no "soul" or consciousness & thus simply be an embodiment of chakra (similar to susanoo)."
But then after yin kurama spoke I had to re-evaluate my conclusion, for a while I actually thought that minato was an honest to god jinchuuriki...But then it hit me, & the answer was quite simple really.
Neither him nor naruto are in fact jinchuuriki, they are simply two parts of the whole definition. Kurama & yin kurama aren't separate bijuu, they are
1 bijuu who is separated, there is a difference.
So in conclusion, minato is not a jinchuuriki & neither is naruto in fact, they are both pseudo jinchuuriki because they only possess pieces of the entire bijuu. Kurama is the living embodiment of, well...kurama. & yin kurama is just the spiritual embodiment, you need both to become complete & thus an actual jinchuuriki.
This also still explains why minato can be an edo jinchuuriki, because he isnt really one
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