Does the density parameter alone determine the fate of the universe?

The first of your two sources is incorrect. The fate of the universe depends on the total density parameter and the contribution of dark energy to that total density.

Universes can be open, closed or flat and these will occur for cases where the total density parameter is $<1$, $1$ or $>1$ respectively. However, this just refers to the geometry of the universe. Only in the case of a universe with no dark energy are these terms synonymous with universes that will expand forever, expand asymptotically to a standstill or eventually recollapse.

The presence of dark energy complicates matters as shown below. This plot, from the supernova cosmology project, shows constraints on the density due to matter and that due to dark energy. The diagonal line marks where the total of the two is unity and we have a flat universe (strongly favoured by cosmic microwave background results). A curved locus is also shown that divides those universes that will expand forever and those that will eventually recollapse.

The thing to notice is that there are areas of parameter space where closed universes can expand forever and also regions where open universes can recollapse (but only if $Omega_{Lambda}<0$).

At present, the best bet is that we live in an almost flat universe that is accelerating and will expand forever.

The reason for this behaviour is that in an expanding universe, the behaviour of the matter and dark energy densities are different. Whilst the matter density decreases as the cube of the scale factor, the dark energy density may remain constant. Thus a universe can only recollapse if it stops it's expansion before dark energy becomes too dominant.