The discussion would be helped if people consulted what Euclid wrote.

The trouble is that in 2-D Euclidean space, there are many equivalent definitions of "parallel"

It would be better to put that as there being many concepts which, in the presence of the 5 postulates, are all equivalent to the one that Euclid calls "parallel". When one is not considering the foundations of geometry, it does not matter which of these properties one calls "parallel", as one understands that when any of these properties is satisfied, all are. When one is considering the foundations, it does matter, and only confusion can result from using any but Euclid's.

But the issue of the 5th postulate is not about definitions. Euclid's 5th postulate does not mention parallelism at all. There are many other 5th postulates one can substitute for Euclid's and get the same geometry, but the problem (so I gather from a few minutes wiki-ing) was that all of them seemed rather more complicated than the other four, leading many mathematicians to search for a proof that would render them all unnecessary.

Bolyai and Lobachevsky (and Gauss before them, but unpublished) settled the matter by working out what looked like a consistent theory of hyperbolic geometry. I say "looked like", because mathematical logic was yet to be invented, and even Hilbert's axioms were still in the future. Models of hyperbolic geometry within Euclidean geometry were found, still in the 19th century, definitively settling the matter.