you don't have any proof that higher DMI means higher skill pops and traning effectiveness because this is not what the second statement says. DMI doesn't affect the effectiveness of training. it's just an indicator of how good the skills of your player are.
to summarise, training positively affects DMI (because it improves the skills) but DMI does not affect training. the DMI level by itself makes no difference to how well the player will train.
if you check the DMI of your trainee when his game shape is highest (level 9, which is the ceiling, thus no sublevels), you will notice that every week his game shape is 9 the DMI will be higher, unless he didn't receive any training (in this case the DMI will be the same)