Agreed with the above, loosing all of a sudden one engine on a twin, for me, its a Mayday, spoken at the earliest opportunity to ATC. Gives me full access to all emergency equipment and priority to do whatever my airline wants me to fly (EO-SID). With a Mayday, now it is ATC's job to empty the airspace for me (I would telling them what my intentions are).
Now, EO-SID is very particular to every airline. I´ve seen 2 different ones when comparing to a friend's flying for another major, same aircraft type, same RWY and airfield. ATC does not have a clue about EO-SIDS as they are, as written at ICAO 8168, the operator's responsibility to check a convenient departure path if something goes wrong.
And then, some airlines will prefer the flight crew fly an easy, straight EO-SID while others prefer to maximise the allowable take off mass and then propose to fly a complicated one. Depends on each airline.