What Jim said.
The only reason you can say you have different phase angles is that you are shifting your reference points. If you select as your reference point one of the terminals connected to the motor, and measure the voltage to the other terminal, you will always have a single 240V phase. The apparent 60 degree phase angle difference that you see when you select the third (not connected) delta terminal as your reference point is not relevant as far as the current through the coil and the magnetic field created by the coil. As far as all of the _intentional_ current paths all of the grounding flavors are the same.
Where you _might_ see a difference is in any circuit path from the coil to the frame of the motor, since the frame of the motor is grounded. The voltage that is trying to 'break down' the winding insulation, leakage through the insulation, capacitive coupling between the coils and the frame, etc. All of these are circuit paths that bypass the _intentional_ current paths. Depending upon the grounding of the delta and the connection of the motor to that delta, you will put different voltages across these parasitic circuits, and thus see different current through them.
For conventional low voltage motors at 60Hz, I doubt that this is an issue, but it certainly is an issue at higher voltages and frequencies.
Also note that in your list of 'delta flavors' you did not mention the option of 'open' versus 'closed' delta. The voltage across the 'open' side of an open delta will be less stable than the voltage across the closed side, and for most open delta services, the voltage across the grounded side will have different stability characteristics than that across the stinger side, because different size transformers are used. This can certainly affect motor performance.
-Jon