As an electrician in a former life, I can guarantee that your motion-sensing switch will not work in a 3-way circuit, unless it says on the box it's designed for a 3-way circuit.
Two of the three wires you are seeing are called "travelers". The red one is probably one of them. A good electrician will have twisted them together. Either one or the other will carry current, depending on the setting of the two switches in the circuit.
The other wire is either a "hot" (line) or a "switch-leg" (load). If you look at the 3-way switch you removed, you will see that two of the screws are silver and the other is brass. The brass screw is attached to the hot or switchleg, depending on which end of the circuit the switch is on. The travelers are attached to the silver screws.
If your motion-detecting switch doesn't have three screws like that (or three wires with appropriate labels), it is not a three-way switch, and will not work in the circuit.
The only way to determine the hot wire is with a circuit tester, with the power on, which is not something I would recommend for a non-electrician.
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