The easiest way to tell which version of The Wrath of Khan you're watching is by listening to the dialogue in the scene early on where "midshipman first class Peter Preston" shows up.
If Scotty says, "My sister's youngest, Admiral. Crazy to get to space," then it's the director's cut.
If that line is absent, then it's the theatrical cut.
If midshipman Preston doesn't show up at all... strange things are afoot.