Yes, dizygotic are fraternal and monozygotic are identical. But that is different than when we're using the terms di/di or mono/di, etc. The zygote is referring to the egg - di = there were two eggs, mono = there was one that split. The term di/di is short for dichorionic, diamniotic. It's referring to the placenta (chorion) and the sack (amnion): di/di = two sacks, two placentas.
All fraternal twins are di/di (although sometimes placentas fuse and look like they're mono/di). 20-30% of identicals are also di/di. A mono/mono is definitely identical, and a mono/di could be either, although I think a skilled tech can tell if the placentas are really single or two that fused, and if they're really single it's definitely identical.
Why do we bother with knowing di/di, mono/di or mono/mono? There are different risks associated with each of the three options. di/di twins are in the lowest risk category. mono/di twins are at risk for TTTS (twin-twin transfusion syndrome), and mono-mono twins are very high risk, since they share one sack they can get tangled in each other's umbilical cords, and they're also at risk for TTTS. di/di twins can still have discordant growth - it's not as serious as TTTS, but that's why they want to monitor your twin's growth with ultrasound regularly.
Yes, they'll be able to tell you at your 20 week u/s. I'm not sure if it's easier to tell if the placentas are fused or single earlier on or not.
Good luck at your u/s! keep us posted.