This. Barrel = ground is the standard; you shouldn't go against this standard unless you have a specific reason.
I thought the two prongs above the "J" was the barrel, and the long rectangle was the center?
On further research, you may be right. I can't tell anymore, that's a horrendous symbol if that's the case -- the rectangle looks so much like a sleeve/shield that slips around the barrel there. I'd never look at that symbol and think the rectangle represents a centre pin; it's rotated 90 degrees from the right place for it to be a centre pin! I much prefer this symbol:
Just to make up for being wrong, I'll point out that pin 2 above (and the corresponding pin in your schematic) is not a barrel pin per se, but a is-there-a-plug-plugged-in-detection-switch. Handy for switching between battery and DC power, for instance. When the plug is plugged in, pin 2 is floating, and pin 1 is connected to the barrel. When the plug is removed, pins 1 and 2 are shorted together.
Let's not get hung up on the schematic though, just make sure your real-life implementation is barrel-ground, centre-positive