No, it's mostly correct. It's not entirely so, as things just aren't as black and white as textbooks say (general hazard of trying to teach facts to young students overall basic facts/ideas that are not grad students or PHd's who can analyze stuff), but yeah--lifting weights doesn't "burn fat" as a fuel in the same way jogging 5 miles at a heart rate of 130 burns fat as fuel.
But lifting weights really hard is usually better.
It's better because it leads to a longer, more sustained metabolic activity that DOES burn fat to use as fuel for tissue repair and strengthening processes and replenishing glycogen stores. You only burn fat while you're jogging. In simplified terms, when you stop, the fat burning stops after your heart rate drops. You burn fat with weight training for up to 2 days after a session because you are still repairing tissue and such.
Basically, it is a logical fallacy to look at things--ESPECIALLY biological processes--in isolation. The "running burns fat and lifting doesn't" crowd might be technically correct--kinda sort of--but they're missing the forest for the trees. What matters most is NOT what fuel you are burning in the 45 minutes you are exercising. What matters is the effect that 45 minute workout has on the rest of your day, the day tomorrow, and all that.
It is more metabolically expensive to repair tissue and glycogen stores as well as strenthen muscle than it is to just burn off a small amount of fat stores, even though in that hour long window you might burn 5x more fat as workout fuel with jogging than you would with weight training.
EDIT--obvious exception for people like Michael Phelps who are spending 6+ hours daily on endurance work. I'm talking about "most people" here.