The old way of doing things was to tell kids they are "smart" or "gifted," which resulted in many of those students simply giving up when they failed because smart and gifted are static qualities. So when a smart or gifted child fails, they simply weren't smart or gifted enough. I know many "gifted" students who never amounted to anything because they failed to work at life and simply expected their "gift" to bring them everything.
Based on this article, the "false growth mindset" appears to be rewarding effort for effort's sake. "Effort" is replacing "gifted" as being associated with inevitable success. Just because you put effort into something doesn't mean you will succeed, and we must be careful not to, as Dweck provides as example, praise a student who failed a test on their "effort." If a student fails, they should be motivated to work harder. I try to do this with my kids. When they fail, I do praise their effort, but I stress the fact that they must now put more effort in to overcome their failures.