All the speculation on best OEI procedures is interesting, but without test or an approved procedure it is all just speculation.
A vertical rise above the deck with a nose down acceleration is the only way to assure avoiding striking the deck edge if an engine quits during the takeoff. How high to go depends on the aircraft, weight and OEI power left, and is not one-height-fits-all.
The most critical time in an elevated deck takeoff is the first few feet, where you most get clear of the platform and netting. A failure there is nearly catastrophic, with a tumble down to the sea likely while just half off the rig. Any procedure that cuts your time there is the best one. We do a vertical procedure in zero exposure operations, which means going up to a height above the pad, then diving to clear the pad lip. If loaded to the wat curve, a failure anywhere during the takeoff results in a landback or a dive away and climb. The critical measurement is tail clearance from the rig during the OEI dive away.
The actual exposure time is only about 5 seconds or so, and the probability of engine failure in that tiny window is truly nil. For the T700 family, the US Army experiences 1 failure each 500,000 hours of operation, so if the 5 seconds of exposure were repeated each hour, and you fly 1000 hours per year, and you have Cat A stay-up ability otherwise, the probability of an engine failure occuring to you during the takeoff exposure window would be about 0.000000000005. Typical rig engine failure experience bears this out.
We debate engine failure, practice for it, work like fiends to polish the procedures, and meanwhile ignore the common fly-into-the-water cause of 1/2 of our fatal mishaps! We are like some ancient religion praying to the engine failure god while each night another beast comes along and steals our best. We don't even recognize his existence!
See the OGP report for a summary of fatal accident causes for twins offshore. 45% is CFIT, 22% Mid air, 22% control problems. Let's debate the real causes, not just the ones we can practice!
http://www.ogp.org.uk/pubs/300.pdf
Nick