Our definition of “invasive” is that it has to be introduced to somewhere outside of its native range, as you've heard, and it has to have potential ecological, social and economic impacts.
For something like the mountain pine beetle, it's spreading further than it ever has before in history. In that case, we would consider it invasive and not just expanding on its native range in a natural process, because some of the factors that led to the spread of the mountain pine beetle weren't natural processes at all. It had to do with the history of fire suppression by humans on the landscape, making the forest much more susceptible to this spread. It has to do with climate change, which can be attributed to human activity. These aren't natural processes that are actually leading to the spread.