Would this be considered manslaughter?

The question goes to intent. If there's a reasonable suspicion that pushing someone like that could lead to their death, then it's more in the realm of murder. If the intent was just to cause injury and there was no foreseeable risk that death could be a result, then it would be manslaughter. The exception to that would be if it was during the commission of another crime. For instance, if someone's robbing a convenience store and shoves the clerk or another customer and causes their death, then it'll likely be murder or potentially a capital murder.

Yes it would. Whatever you intended, you caused death so it has to be at least manslaughter.

I'm British and this would be referred to as a "one punch" case. Fairly controversial but for the moment it's considered as manslaughter.

I was in prison a few years ago on the same wing as another prisoner convicted of this and I found out what he did when I was helping him with an essay for the education course we were both doing. (Hehe - easiest help I ever gave, and I had done a lot of it because it was my paid prison job to help out in class. I was ready with all the usual advice about a beginning, a middle and an end, but all he really needed was something to start him off on the topic. We chatted about the topic, we hit on something he could relate to from personal experience, and that was it - his pen was off and flying. He doesn't need ME now, so I made some tea! He stayed in my cell to write it so I could read it afterwards and it was good. I know he's not stupid, but just sometimes when you get set an essay to write it's hard to get started, that's all.)

Anyway... he got into an argument, hit the other man, and the other man fell down, hit his head and died. This resulted in a manslaughter conviction. He's a tall big guy and I can easily imagine that if he got angry and lashed out with a punch. He also told me he has Asperger's Syndrome and that made sense - obviously in him it's mild, but it can still result in over-reacting if he gets annoyed.

Of course. But it's not manslaughter. Manslaughter is when it's not intentional. Deliberately pushing or tripping someone knowing they will fall to their death is homicide.