The easiest way for me to remember is that DZ twins are always dichorionic/diamniotic. They come from 2 eggs, so there's always 2 chorions and 2 amnions (one for each egg).
When it comes to MZ twins, it's a little more complicated, but I really like this picture
for explaining it.
It says that among MZ twins, mono/di is the most common, and I'm not sure if that's true. But that is what mine were.
MZ mono/mono is very rare and the most difficult for a pregnancy because of the risk of cord entanglement.
If twins are di/di and the same sex, there's no way to know if they're MZ or DZ until they're born. And even then, some DZ twins can look really alike.
As far as "riskiness" of pregnancy, I'd feel comfortable homebirthing with di/di twins (either MZ or DZ); MZ twins who are mono/di are at risk for TTTS so I'd maybe homebirth only if I were also seeing a Ob regularly for TTTS checks; and MZ mono/mono are very risky and usually delivered by scheduled c-s around 32w (or earlier) because of the risks I mentioned.
Someone correct me if I'm wrong; I used to know this stuff really well but now my twins are almost 8!