In my time the only 4 engine aircraft we had were DC-8's. I never bid those and didn't fly them, so I don't recall their specs.
I did fly the L-1011, and "nearest suitable" was standard for that, although there could be extenuating or mitigating circumstances where that was not the case. This usually involved "Captain's discretion". Same for the two motor aircraft.
I can say that in our "culture" and I believe this is true for all US airlines, you're not going to "coast out" with an engine inoperative. You most certainly are not continuing to a long-range destination after shutting one down on takeoff.
On short-haul stuff, Birmingham, AL to Atlanta, GA.. yeah, if you lost an engine out of Birmingham, you'd probably go to ATL. By the time you got cleaned up and the checklists run, you'd be to ATL anyway and ATL is THE maintenance base.
But you're not going from LAX to ATL after losing one on takeoff. You most surely are not going LGW to ATL after losing one on takeoff.
Still, I can understand a different "culture". I'm sure in the BOAC days it was far better to press on home than land back at the jungle airstrip where they were filtering the gas through a chamois.
Stuff like that gets ingrained in the corporate culture.
Yeah, ETOPS are a different case and the procedures are pretty well nailed down and "boilerplate". Still, in an ETOPS aircraft you sure as heck aren't going to even enter the tracks with an engine failure. An ETOPS aircraft losing one on takeoff isn't going anywhere.