Hello,
I worked for several companies and in one company the name for "engine failure procedure" was EOSID (Engine Out SID) in another it was simple EFP (Engine failure procedure) another called it EOP (Engine Out Procedure)... but it's all the same.
I think Jeppesen called it EFP while Lido called it EOSID.
Since on our airplane with an engine out we'll never meet the required gradient we always have an engine failure procedure and if there is nothing specified it's simple straight ahead.
The most important is that the operator is responsible to establish such procedures when you are not able to meet the climb gradients of a normal SID in case of an engine failure. So ATC doesn't know your engine out procedure! For this reason it's very important to report the procedure to ATC!