Are you planing on getting a wood lathe or a machine lathe?

Not sure what the difference is between those. I'm looking at this
https://www.amazon.com/gp/product/B000UCK0W6/

I know it's cheap, I know it'll have problems and suck, but that's all fine because I can always buy a better one later if I find out it's actually a useful tool for me.
I don't have much intent other than being able to shape some various not too hard woods.

Ah ok. You can make plenty of things! You can make tool handles, plates, decorations (because why not) and things of the sort! My father works with it all the time because he works on powder horns for reenactments of war of 1812, american revolution and so on. He uses it to make end caps for the horns so that they are pretty well sealed, but he has also been able to make tooth pick holders with it so there are plenty of options of things to make with a lathe.

As far as I know about the difference between a machine lathe and a wood lathe, a machine lathe can also turn metal if need be, but nothing much else is between the two tbh. :P

You can make a baseball bat, a rocket engine nozzle (if able to work in metal), various other cylindrical or spherical objects.

Not really sure as to specific things, it really just comes down to what you wanna make really. 'w';; I don't have one myself though so I don't fully know what can be done with them but I know there's a fair amount that can be made on one (I wanna get one myself one day though cuz I wanna make ROCKET ENGINES >:V).

It's gonna be a cheap lathe if I get one, probably only good for stuff like wood. I'd still try to make a rocket out of it though and if it blows up that's still a fun result. I'd use crushed up sparklers as a fuel because that stuff burns sooo bright :D