First Extreme – "You should crush yourself every workout."
This is dumb. The idea that "if some is good, more must be better" pervades the meathead community. A lot of lifters feel the need to beat the shit out of themselves day in and day out. In fact, most strong people go through a phase in their first few years of training where they treat every workout like it's their last and they refuse to take time off out of fear that they'll lose all their progress.
I've been there myself and I actually think that "overwork" phase of a lifter's career is invaluable for instilling a good work ethic. But sooner or later, all strong lifters realize – either due to injury, burnout, or just listening to the advice of more experienced lifters – that you can't go balls to the wall all the time or your progress will stall. Worse, you'll get hurt.
Sometimes when life gets busy and stress is high, taking a break or at least a step back is the best choice you can make to keep the gains coming.
Second Extreme – "When you leave the gym you should feel better than when you came in."
This sentiment probably arose to combat the idea of people judging the efficacy of their workout based on how close they come to death. I get that, and the idea is well intentioned, but it's going to the opposite extreme.
Unless you're just doing some stretching and foam rolling, you're not actually going to leave the gym feeling better than when you came in. If you do, then you need to be training harder. After all, it's a gym, not a spa where they heap hot stones on your tush.
I understand the backlash against workout maniacs who habitually try to kill themselves, but you still have to train hard and push yourself... unless of course you're cool with being mediocre. A good workout should leave you temporarily gassed, but it shouldn't leave you crippled or crushed for days.
Like most things in life, the answer lies somewhere between the extremes.