There's a fundamental problem with this question that makes it unanswerable in real physical terms.  To understand why, we need to think about physics for a moment and how mathematical models are not reality ([the map is not the territory][1].)

Let's start with some basic concepts.  Movement is a change of location. Movement inherently requires the passage of time.  Speed is (in this context) the rate of movement over time.  So, when we say something is moving or has moved, what we mean is that at time A, an object was at a certain location and at time B, it was at some other location.  So, another way to ask your question is "at what time did the speed of the car go from 0 to something more than 0?"

It's fine to come up with logical arguments about this and there's a long history of that.  But physics is not just logical arguments and math problems. What distinguishes physics (and science in general) is empirical observations.

How can we experimentally determine when this movement starts?  One option would be to set up a video camera with a known frame rate.  We can examine the frames and at find the first one where the vehicle's position has changed.  Great. But there's a problem: we only know which interval the movement started.  But during that interval, we have the exact same problem as we started with.  So, we get a faster camera and shorten the interval.  We get a faster camera.  We use a laser.  We keep making the interval smaller.  Ultimately, we hit the limits of precision of our technology.  No matter how precise the technology is, though, we will still have an interval, not a moment.

And mainstream physics says this isn't just a limit of our current technology, but rather a fundamental aspect of nature.  There's simply no way to take a measurement without some finite interval of time.  The best we will ever be able to say is that at time A it was at position X and at time B it was at position Y.  What happens during that smallest measurable interval of time is conjecture.


  [1]: https://en.wikipedia.org/wiki/Map%E2%80%93territory_relation