On the last point about traveling by car, If you use Google maps and have the location history turned on. It will show dates and times and mode of transport used. So could hopefully be used to disprove train travel.
I use Google maps and its location facilities constantly, and its "mode of transport" unfortunately doesn't really prove or disprove anything, it is just a best (mostly good) guess.
True, often it's deadly accurate even for a complex multi-leg journey. But other times, it can mistake a simple, obvious train journey for a car, a bus, a walk and (on one occasion where there was a large lake nearby), a ferry! A few weeks ago it started wrongly labelling *all* my train journeys as car journeys, then after a few days it went back to being mostly accurate.
I go through and manually correct it, to give a reasonable record of my travels. And as an aside, when I've edited my journey, there doesn't seem to be any way to determine that it's not the "original" journey, as recorded by Google.
The actual location can occasionally be out by 10-20 miles; and it pretty much stops working if you're in low battery mode.
Some of the accuracy no doubt depends on how good your phone is, but it's always going to be guessing, particularly when a train and road journey follow broadly similar routes and take similar times.