So after acquiring confirmation about the Two-Base Theory here is my idea how to link everything together:
Goku during DBZ had Base, SS1, SS2, SS3.
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Then he and the other Earth Saiyans did the ritual and he became Super Saiyan God.
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During his fight with Beerus he turns back into a Super Saiyan, but is still just as strong. Beerus explains that the power merged with Goku. I would modify this and say that Beerus is slightly wrong ... the power only partially merged with his being. This is why we only see the God-like Super Saiyan once. ... The power wasn't done merging.
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Goku fights Beerus one last time and their attacks create a sun ball that is heading towards Earth. In order to stop it Goku needs more power. During his fight, he turns from a Super Saiyan into Base form. Surprisingly, this change to a weaker form gives him enough power to defeat the sun ball. That's because the SSGod power fully merged with him in that moment. He now only has a powerful Base.
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Goku meets up with Whis and Vegeta (who accessed the red-haired god form on his own. Vegeta then had the power partially merge into his Super Saiyan form and then fully merge into his Base form like Goku). In order to gain more power and control, Whis teaches them to not let their ki leak out. They go into the Staff Dimension and achieve SSBlue. Now the two of them have Merged SSGod Base and SSBlue when they fight Frieza.
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After defeating Frieza, Goku and Vegeta learn of the other universes. They are recruited to be part of the U7 team. In order to prepare for this, they train in the RoSaT. During this time, Goku and Vegeta master ki control to an even greater level, allowing them to access mortal forms such as Base, SS1, SS2, SS3 in addition to New SSG and SSBlue.
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After the tournament they retain all of these forms. It can be said that Goku fought Botamo, Frost, #17, Krillin, and Bergamo in Base form. He fought Beerus' Sun Ball, Hit, RoF Final Form Frieza, Copy-Vegeta, Monaka-Beerus, Slim Buu in black-haired SSG form. We can make this conclusion because Goku turns into SS1 against Krillin, #17, Frost and Bergamo, which means he wasn't using God-like power. He turned SSBlue against Hit, RoF Frieza, and Copy-Vegeta, so he was using New SSG. Monaka-Beerus and Slim Buu make sense to be New SSG since both of those characters are strong and he could not handle them in Base alone.
Goku Black must have been using New SSG but didn't have full access to it. I believe it was stated in the show that he gained more of Goku's power the longer he fought.
When Vegito appeared in the Future Trunks arc, he must have used New SSG just before he turned SSBlue. Copy-Vegeta must also have been fighting in New SSG, as Base could not beat SS3 Gotenks.
I think this solves everything, basically.