Edit: I guess I should have made it clear this is all head canon and just me trying to give logic to all the forms. I didn't dig deep into all the details so forgive me for any mistakes.
This all started when I learned that Goku's silver hair Ultra Instinct form wasn't true Ultra Instinct, but his body making a form to use/handle it, and UI is just a technique.
So let's start with basic Super Saiyan. It gives a 50x boost. I'm gonna say saiyans have a natural ability to boost their ki, and the golden effect is from the excess ki leaking out. Even once they remove the leak, the hair still reflects the increased pressure. This makes me think Saiyan hair is more connected to their biology than ours, and has ki channels making it more of an organ. They can then force their SSJ multiplier even further in an unnatural way, this gives us SSJ2 and SSJ3. The body is forced to mutate to handle these making them not a natural state and exhausting, this being most noticeable in SSJ3.
Next the god forms. SSG the red form is actually just base form with god ki running through it, however with not being used to the intensity of this ki it is hard to control and reflected in the hair. Then SSGSS the blue form is just using the SSJ multiplier with god ki. In theory they could go SSGSS 2 or 3, but those forms would be impractical for the same reason Goku and Vegeta use base SSJ even after getting SSJ2.
Super Saiyan Rosé is where I started to have fun. I was trying to come up with a reason why it was the pinkish color. What I came up with is that Zamasu is used to using god ki, so there is no leak when he does. His hair remains black even though he is using a higher grade fuel because he is perfectly managing it internally. However when he activates SSJ the form does have a leak, but the leak is only to the red color level instead of reaching the blue levels.
Then I tried to figure out SSJ4 in Daima. It also falls into the reddish color range, however it's not a god ki form. It's a SSJ form seeing as how Goku was in SSJ3 before entering. My theory for this is SSJ3 was the max a Saiyan can push their normal ki output with their base body. However SSJ4 uses more monkey features so I'm thinking it adds the Oozaru form in some way to increase durability. Perhaps the fur acts as a shell holding ki in. This turns the body into a pressure cooker for ki. The user pushes the ki multiplier inside their body till it starts to condense into something like god ki, making the fur and hair turn a reddish color. The form could also be a mix of Broly's Wrathful form mixed with SSJ. Funny enough SSJ3 x Wrathful would make a 4000 x multiplier, getting close to the 5000 of SSG.
Next let's look at Broly. He has his Wrathful form which is just a condensed Oozaru. Then he has his form where his hair spikes up, but isn't gold, we could say this is him using SSJ on top of Wrathful, the increased defense from the Oozaru part maybe keeps the ki contained. It then goes even higher till the body can't contain it and the hair goes gold from the leak. However he eventually gets the green color. This is because he isn't boosting the power of his ki, outside of using SSJ, he is just multiplying the amount. Perhaps this reaction starts to happen outside the body when the leak gets too intense and that causes the green tint. We could say SSJ4 comes from multiplying the power of the ki while keeping it pressurised, but Broly doesn't multiply the power, just the amount.
Ultra Ego is pretty easy to explain, it's just the body mutating to handle Destruction Ki instead of regular or god ki. Once Vegeta masters it he may just look like his base form with purple hair. He could then go SSJ on top of that getting a new color.
Super Saiyan Rage and Gohan Beast could just be explained off as hybrid forms, but I'll throw some ideas out anyway.
Trunks is seen taking energy and channeling it into his sword, so perhaps he has a knack for absorbing energy and using it, the sword of hope for example. Everyone around him was throwing around god ki, so perhaps he started to pull that in, giving the form the blue glow. This also opens up the idea of using two types of ki at once, which I think is pretty neat.
For Gohan Beast, let's say when he got his full potential unlocked it acted similar to Zamasu using god ki. It lets Gohan fully use his super saiyan power level without any leak. He is basically using his normal ki at SSJ boosted levels without entering SSJ, and using it as well as the kai use their god ki. He isn't activating the SSJ power up at all, but instead reaching that power at his human base. Then when he uses Beast it's the SSJ transformation triggering on top of this enhanced base state.
I know there probably wasn't this much thought put into any of the forms, but I like making sense of things in my head. I hope this was at least interesting.