In my experience if runway 27R was not an active runway, it would not be under the control of the tower controller. Instead it would be under the control of the ground controller and be treated as a taxiway. You would have then been told something like this:
"N12345 turn right on E and contact ground .9"
If 27R is an active runway, then 27R would be under the tower controller. If there's no landing or departing traffic on 27R at the time, he can tell you:
"right on E, cross 27R, contact ground .9"
If I were the tower controller I would added a little more to that line to avoid confusion:
"N12345, right on E, cross 27R and contact ground .9 on the other side"
So for me, if the tower guy tells me what you said, I would take that as contact ground after crossing 27R. If the tower guy tells you to cross a runway, then he has control of that runway, not the ground controller. If another aircraft were told to line up and wait but instead started their takeoff roll, then I don't think the tower guy would want you on the ground freq if he needed to suddenly stop you short of 27R.
But then, of course, if there's any doubt, you could always just ask if you should "stay with him" or "switch to ground now".