Hard to tell from the video - but that does sound a lot like piston slap or even rod knock. Especially when it gets louder with off-idle speeds (accelerating) and when the engine is cold. Does it do this when you are in park/neutral? Or only under load?
There are a number of other possibilities that might cause that sort of noise - from a cracked flexplate/transaxle issue (loose bolt making contact with engine side) to worn suspension parts (powertrain was binding).
Exhaust leak is a possibility, but to me, that would sound a little bit different. If you notice a change in exhaust tone if you drive through a puddle, that might put more weight toward an exhaust leak. I have a fairly decent exhaust leak on the Corolla - doesn't quite make the same sound as yours, just sounds like a louder exhaust.
2002 Corolla S, Silverstream, 1.8L 1ZZ-FE (VVT-i)
2003 Matrix XRS, Cosmic Blue, 1.8L 2ZZ-GE (VVTL-i) - RIP 7/14/2010
2009 Matrix XRS, Blue Streak, 2.4L 2AZ-FE (VVT-i)
2009 Toyota RAV4 Limited, Black, 2.5L 2AR-FE (Dual VVT-i)