Here's my two cents:
Main question -- visual up to MSA? If so, and deemed that you have adequate climb on all engines and room to maneuver in case of engine failure, ok to turn as early as you deem safe.
If not visual, follow SID/climb procedure as appropriate up to MSA/COT (climb on track altitude). Unless cleared "when ready direct...", inform ATC that you need to follow SID/CLP up to xxx altitude before turning on course.