I'm just another learner, so maybe a native speaker will come along with better guidance here, but I'm pretty sure that in most cases Swedes would say träningen for the (physical) exercise and övningen for the (mental) exercise. They also have another word, motion, which I think is used for the more general concept of physical exercise, e.g. motion är bra (exercise is good).
Thanks for the clarifications on övningen, träningen, och motion.
Years ago, someone in Uppsala told me I should say "Jag vill öva min svenska" to tell a local "I'd like to practise my Swedish", since cashiers, baristas, etc would switch to English on hearing me speak. Is that a correct use of öva?
You can, but it's not the same as övningen är inte svår. En övning is an exercise or a drill. The Swedish sentence above refers strictly to one session of something (or one specific exercise). If you want to speak about what it's like to do such exercises in general, you would have to say e.g. Det är inte svårt att öva.
Good translations for The training is not hard would be Utbildningen är inte svår or Träningen är inte svår. In these cases, the meaning is much more general, it won't be restricted to the one time as it is with övningen.
In the previous sentence, 'Han behover mer övning', I thought 'övning' was a verb. Something like 'He needs to exercise more' or 'He needs more exercising'.
But here, I see 'en övning' is 'an exercise' and 'öva' is the verb 'to exercise, to practise'. So, how would it be 'He needs to exercise more?' (Google translator says 'Han behöver utöva mer') :/