Yes, but not for Zoan. Zoan awakening was explained the first in ID.
And yes we don't know about other types and I believe oda will just do random stuff...But we can see pattern Logia obviously effects everything around them, and the two paramecia awakening which I mentioned both work a bit like logia(katakuri pretty much is a logia) and DD generates threads(even inside his bodies also seems a bit like logia apart from tangibility).
But from Kid and Law => This was obviously oda just adding "cooler" attacks and tag it as awakening...So there are no rules...and with the zoan awakening its even more broken as earlier some characteristics were mentioned for zoan.
Nah this perfectly explains that too.
Think about it. Those were awakened zoan users, except their zoan powers were extremely base. Minatouros was very clearly just a cow. The others were a rhinoceros, zebra, koala and chihuahua.
Now think about the dialogue this chapter, they talk about how Zoans have a "mind" of their own. Being simple animals, their minds are "simple", hence the snot dripping from them. They were all dumb with high recovery, that was it.
I'd even argue that ancient zoans would not end up super well, because they are simply dinosaurs.
Now think about the potential of Mythological zoans. Sengoku, Kaido, Marco and Luffy all seemly have deity type mythological fruits. Sengoku literally is a giant Buddha. Marco is a Phoenix. Kaido is a seiryu. Luffy's is made up but Nika has been referred to as a sun god.
Basically, zoan awakening probably gives these people different personalities, but doesn't make the user dumb. Hence, Kaido potentially has an amazing awakening coming, the spirit of a dragon