Q: Drove without oil; car shaking, cut out

If the engine had no oil in it, the shaking, noise, and eventual seizure of the engine would have been the result of that. The engine is also more likely to overheat. The white smoke that you saw is likely coolant that has blown out of the radiator cap or the top of the reservoir due to over expansion. This coolant likely landed on something very hot such as the exhaust or the overheated engine block causing it to burn off quickly. I would first recommend having your engine inspected for damages by a qualified technician. Best case scenario would be adding engine oil to the correct level as well as coolant and starting the engine and it runs fine. If that is not the case, permanent engine damage may have occurred.