The moving while stationary thing has been a problem for as long as GPS has existed. The math is complicated just as much as the reasons are for the problem - there are a long list of factors that are difficult to eliminate, so every measurement will have error.

Some units will go in to averaging mode when they think you are stationary - convolving all the points in to an ellipse of error with you in the middle more or less at zero kph, others just give it to you raw. If you track this slow movement you'll see the points all roughly surround where you actually are.

It's not a bug, it can't be fixed, it's just a result of some amazing human ingenuity, the accuracy of it is astounding. Though I do understand when you feel it isn't :-)