If you watch clips or read through the official trade postings from the NHL, even conditional picks are defined by which team it originally belonged to.
The only exceptions I can think of are when a team owns multiple picks in a round at the time of a trade, and the conditions are team A gets the higher of the multiple picks.
an example from the Patrick Eaves trade:
"*Conditions- Based on final draft position, Dallas will acquire the middle pick of the 3 picks Toronto had acquired, one of which was sent to Anaheim in a previous trade. The selection will be one from the 2017 draft (Ottawa 2nd round pick, San Jose 2nd round pick and Toronto 2nd round pick).
If Eaves plays in 50% of the games in first two rounds and Anaheim advances to the third round, the selection becomes Anaheim's 2017 1st round pick."
Even then the options of the conditional pick were limited to the 3 picks Toronto originally had at the time of the trade.