Segwit: dropping signature from relay
Yes and no: this is a presentation from 2015, and you are referencing a section which Pieter described as "assume we can redesign bitcoin". So bitcoin hasn't been redesigned, and not everything has been implemented as described in 6:20. 

How it is handled today is part of a discussion with Andrew in bitcointalk (read down to page end). Older nodes don't get to see the witness data. They get just tx data, in an "anyone can spend" way, so it is recognized as valid tx by older nodes. Newer nodes get the witness data (of course). 


  Am i right saying that signatures for segwit outputs will never be
  relayed to old clients?


yes.