You probably don't realize the differences between the latter 3 because there aren't any differences per se. Well, there are a host of similar techniques that go by various names, and they all get confused with each other. First of all, real "displacement mapping" means actually changing the geometry, which means if you want it to look reasonably good, then your geometry density had better be damn good in relation to your map resolution.
Parallax mapping is the more generic name if you go by the literature, but it's also called offset mapping by some. It's simply the idea of transforming the view vector into tangent space and using that as the basic direction along which you change texture coordinates according to a height map value. So (depending on the direction of offset) you basically take height map value * some scale / TangentView.Z = some value, and some value * TangentView.XY * scale factor relating to texture size = the UV offset. If you use it in combination with bump/normal mapping, it might be called offset-bump or 3d-bump especially, but I don't think that's really necessary since it's essentially a precondition for all texture-based operations including normal mapping.
Some people will also take into account that once you've offset the texture coordinates, the height in the height map is different from what you originally sampled, so they repeat the process a few times (doing fixed-point iteration) or do some ray stepping along the view vector to get a more "correct" offset position by determining where the view vector intersects the height map, and this might be called "steep parallax mapping" or "relief mapping" depending on what they do to get that result, and the results are certainly better, but it's obviously a lot more costly, and so you have to question whether it's worth it or not.
I've often found in practice that simple parallax mapping or a few (3 or 4) fixed-point iterations plus a little bit of extra geometry to accentuate shape (assuming that you're not trying to simulate small variations on an otherwise flat surface, but something that has some substantial shape to it) often gives the most convincing results.