I often hear that explanation of the rule, often from players that have been playing the game a long time. It's almost part of hockey lore. Many refs at the amateur level even think it's a rule.
Except that the rule doesn't exist. Rule 91 defines tripping as:
"A minor penalty shall be imposed on any player who shall place his stick or any portion of his body in such a manner that it shall cause his opponent to trip and fall."
The closest it comes to saying that if you touch the puck first it's not a trip is NOTE 2:
"If, in the opinion of the Referee, a player is unquestionably hook-checking the puck and obtains possession of it, thereby tripping the puck carrier, no penalty shall be imposed."
Touching the puck doesn't give you carte blanche to take your opponent's legs out from under them. You have to actually get control of the puck and have the trip occur incidentally, not just knock the puck away while you check at the knee. The rule specifically does not mention poke checking.
If all you had to do was touch the puck first, why wouldn't every play consist of touching the puck and then taking the other player's legs out from under them with a trip? That's a lot easier than having to body someone off the puck.
In the case of the Cole takedown, clearly the Leaf player didn't get control of the puck, he just knocked it away. And he shoulder checked at the knee, which frankly I don't see how you can interpret as anything other than intent to injure.