You know what, it's just the context difference (for me)
На поїзді <~> where? and поїздом <~> how?
- Де ти зараз? Що робиш? - Я їду на поїзді. (Where are you now? What are you doing? - I'm on a train)
- Як ти будеш туди їхати? - Я їду поїздом. (How will you get there? - I'm going by train)
Note: even the names of the cases represent this difference:
На поїзді - місцевий (Locativ), for locations
Поїздом - орудний (Instrumentalis), for tools and using things
Your tutor laughs because "на" is grammatically incorrect, unless it's a bike. If you come by bus - you come "автобусом". if you are on the bus - you are "в автобусі". Same for Plane, Train, Tram, Car.. it's possible to SAY "на автобусі", people will understand you (lot of them even say like this and think it's correct), BUT it means that you are on the roof of the bus (although, in some Asian countries people travel literally ON a bus/train)