(It's tuxedo, not taxedo)
This is just more of the same old arguement. Just because you can get away with it, does NOT make it right. I can say nothing about Princess, but I can tell you what I saw on Voyager just a month ago: Lots of folks were in tuxes, those that weren't were either in suits or sport coats. I don't think there was any significant percentage that didn't put on a tie. I can assure you, nobody was wearing shorts and CERTAINLY nobody sane would go to the trouble of wearing a tux on top, but shorts below (where's the benefit - other than being silliy on purpose?).
I can assure you that were somebody to appear in such an outfit AND be let into the dining room (a dubious possiblity) he would not be "everyone's favorite." Many, many people would find it rediculous. He WOULD be everyone's clown. Is that something to aim for? Do you enjoy having numerous people laugh about you behind your back? Some people do. That's your decision.
Maybe not EVERYONE cares if you go extra-casual on formal nights, but most people do. What you do on the ship is one thing, but in the formal dining room on formal night is different.
Look, this is your first cruise. Your mother is either giving you bad information, or bad advice. There are a lot of things you can do that you will get away with (at least until cruise lines start enforcing their own policies), but is that really what you want? You should not base your life on doing the minimum that you can get away with - even if some choose to do so.
Take my advice, bring proper attire for your first cruise. If you find that I'm lying to you and half of the people at wearing casual wear at formal night diners and that those who dress most clownishly are the most respected among everyone, then I'll eat my keyboard. Most likely, you will find that most people are dressed elegantly and that the 10% of adults that refuse to do so look out of place.