Your ideas are logically correct.
But to "join in" is an idiom, according to thefreedictionary.com, which has been useful to me, as a native English speaker, to pin down precise idiom meanings. "Join in" assumes intent or choice. "Join into" is more for strictly physically joining.
- "We're eating ice cream. Would you like to join in?"
- My friends, who were racing to the park, let me join in.
- I joined my friends in eating ice cream.
- I joined my friends in racing to the park.
However, as you mentioned earlier, "join into" can be appropriate for physical joining, especially of two non-person subjects. This is different from "join in" where someone starts to act the same as others.
- The Missouri River joins into the Mississippi River north of St. Louis.
- East Germany was joined into West Germany in 1990.
Also, both "The two sides were joined into battle" or "The two sides were joined in battle" sound right to me, although "in" seems a bit better. This is an edge case I can't explain precisely why, but my rough explanation is that they could have done so intentionally or unintentionally, and they're more likely to have done so intentionally. But they could have been, say, forced to the same field by the weather.
Incidentally, the extra subject between join and in doesn't matter here, although you could change
"I joined my family in cooking dinner" with "I joined in with my family cooking dinner."
Though I prefer join (subject) in.
I think there may be another way to look at it, but the main things are:
- join in is an idiom
- join into is more strictly physical, without intent. For instance, in the above phrase, "The commoners joined the king's army in taunting the rebels." also works, but it involves less physical activity. So maybe you can also see if you can replace the more active motion verb with something less active, and if so, "in" is preferred to "into."