Yes, I realize the western horse is not banded but it was a quickly found picture depicting two different disciplines.
My take on this is what your trainer told you to some degree. Braid if your horse is more hunter built; band if more western built. If you braid and show in halter and showmanship, please match your attire accordingly. Use a bridle if braided and a well fitted halter if banded for the halter and showmanship classes.
As for forgiving a less than perfect lope on a braided horse? Don't bet on it. If it's a pleasure class, it will be judged per pleasure standards and braids and bands won't matter. If it's a performance class such as hunter under saddle or western pleasure, again, it will be judged on the acceptable standards and the mane should have no affect on your placing. I say 'should' because there may undoubtedly be someone judging a show that could be biased to the turnout and weight it much heavier than I would. If you've braided or banded, most likely the rest of the turnout is going to up to showing standards so the style of the mane would not sway me one way or another.
The two links below are great, short tutorials on the subject of banding and braiding. I suggest you practice a lot. It's not hard to make great braid buttons or band but does take some practice to get it right.
How to braid a mane
How to band a mane