This is a subtle thing - hard on means something had effects that were difficult for the person to deal with, or which had a negative impact on them at least. It was hard for him means that it was a difficult process, something he had trouble with.
Now those two meanings have a lot of crossover, so you can often use them interchangeably, but I assume the person you were talking to didn't like the implication that the cousin found the training hard - as in, he had trouble doing it, he wasn't very good at it, he wasn't strong or fit enough etc.
You could argue that if it was hard on him then he was still struggling for one reason or another, and he wasn't able to cope with it all easily... but it was hard on him sounds a little more like "it was a tough situation, but he got through it" whereas it was hard for him sounds more like "it was hard because he wasn't very good".
It's subtle, but people can be proud about these things - so you're not wrong, it's just that sometimes you need to be careful with how you phrase things, and this is probably one of those situations. Honestly nobody should really expect a non-native speaker to have that level of finesse, so the fact they corrected you implies (to me) that their feelings might have been hurt, or they're sensitive about it at least!