Everything we're seeing in the p4 is point to ACN being the required sub-class of BLM. The info you linked is based off the old 1.0 info.
When the Limit Break bar at the top fills up, it will start flashing Yellow. When that happens, if you click the action button Limit Break, you'll get a targeting reticule. Click that where you want the LB to go off, and after a short cast time.. Boom.
Once the first bar fills, a second bar will appear and start to fill as well. This can happen 3 times.
Ive only seen the lvl 1 LB as most are to spam happy to let it get to lvl 2 or lvl 3.