While the you pull it, you tank it has been a saying for a while, its getting a bit twisted due to being shortened. As "if you pull it, expect to tank it" comes more from the earlier MMOs when it was sometimes better for the party as a whole to just let whomever accidentally or intentionally pulled an extra mob or mobs to die with no assistance in order to prevent a party wipe.
Pulling of mobs is the Puller's responsibility. Not the dps, not the tank, and not the healer. As the Puller has to consider everything that has already been mentioned in the thread, in addition to their normal job functions. Just in more modern MMOs Puller and Tank are considered one and the same to many.
You could really pull on any class and do fine depending on how you do it. Like if a dps, and think things are going slow but tank and healer are having issues with big pulls? If you know the dungeon so well try focusing down the biggest threats, and go get the next mobs when the minor trash is close to dead to keep things flowing. It will probably be of more help in getting newer players used to bigger pulls without forcing wipes and upset party members.