It is only in the absence of dark energy that the correspondence between geometrical curvature and the ultimate fate of the universe is as straightforward as you describe.
Measurements (primarily of the cosmic microwave background) indicate that our universe is flat or very nearly so, which should be interpreted geometrically (i.e. in terms of the sum of the angles of a geodesic triangle). In the absence of dark energy, this would correspond to a scenario in which the universe continued to expand but asymptotically approached zero expansion velocity.
However, the concurrent measurements of the presence dark energy suggest that our geometrically flat universe will continue to undergo accelerated expansion.
The influence of dark energy is sometimes neglected in popular accounts, leading to much confusion among non-experts. Keep in mind, though, that it was only in the last 15 years or so that scientists had any direct evidence for the presence of dark energy, so it might be understandable why it was put aside in earlier simplified explanations.
A discussion of the ultimate fate of the universe, its connection to geometrical curvature, and the role of dark energy, is found in the wiki article.