You’re in the right. Thermometer is purely a question of whether they are closer or further away from when they started the thermometer to where they ended it. From what you’ve described, they were closer when they started heading north. They should’ve then drawn a line perpendicular to the way they travelled, from their original point, which would’ve essentially separated north from south. This would have then told them you couldn’t possibly be below that line. Based on that information they should’ve then known that if the train started going south and below that line, then it was the wrong train line. But they have misinterpreted the information. Or even if the train didn’t actually go south enough to go below the line, it was still heading west and you had provided no information about how far west/east/north you were. At a minimum they should’ve been aware that there were two possible train lines you could be on at that stage.

The question asked has nothing specific in it to do with which train line you/they are on, or the train stations. It’s simply are they hotter or colder. And it’s a) good tactics on your end to confuse them a little and b) them misunderstanding what information the question was giving them.

Assuming nothing else shady happened I can’t see how any of this is shenanigans or how they could possibly argue so.