Asalamu Alaikum,
Prophet (sws) used to pray 8... and then 3 witr to make it a total of 11 rakaah.
(Which is recommended... because you read Quran slowly and contemplate on its verses)
Omar (rad) is the one who started praying 20 rakaah.
He's one of the rightly guided Khalifahs... so they are both ok to follow.
Finishing the Quran in taraweeh in Ramadan month... I don't know whether Omar did that or the prophet... or if its something totally new.
But whatever the case.. I think its better to recite Quran slowly during taraweeh than to rush everything just to finish Quran in 30 days.
As far as people leaving in middle... perhaps they are following the prophet sunnah.. or they are just tired... or have to go to work... taraweeh is sunnah, so there is no sin on anyone if they decide to leave early.
However it is said, that it is best to stay with the Imaam till he finishes with witr... whether he does 8 or 20. If you wish to continue prayer you can 2 nafil prayer for the rest of the night... as much as you want.. (especially on the last 10 days of Ramadan)
W'salaamz,
Hamid