Even if determinism was false [..] would this actually imply that the way events turned out could have happened otherwise?
The most common interpretation would be that yes, in that case events could have happened in different ways, and only happened the way they happened due to random factors.
But philosophically it depends on the way that determinism "was false" (or how "determinism" is defined in the question). Causal determinism is the most common meaning of "determinism". Causal determinism can be false in several ways which would still make some or all events determined by something.
Even if there were true sources of non-determinism in this universe, those could be too small, too few, too rare, too far away or too "regular" to make events turn out in different ways.
Predeterminism and fatalism are alternatives which do not necessarily require determinism. There can be more such similar nuanced theories that make events non-random yet not decided by the previous state either (but by anything else). All those could be called "Determinism without Causality".
So "events could not have turned out a different way" could be the definition of "causal and non-causal determinism". Then even if causal determinism was false, it is logically possible that non-causal determinism remains true.
Time-travelling in the science-fiction sense is one way of introducing non-causality into a universe, just to give a popular example, same as supernatural telekinesis, teleportation and such. Similarly, prophecy or otherwise "seeing" the future without inferring it from the present as in fantasy and mythology would introduce non-causality. Those could still mean that "things could not have turned out otherwise".
In physics, only quantum phenomena are considered to be possibly non-deterministic in a strict sense, even if many other events can only be described stochastically due to lack of precise measurement methods.