Note that not all of the examples you provided are simple letterpress. The second and the latter are hot stamped, which means that a thin metal sheet is being lightly melted into paper fibers through heat and engraved zync plates, which usually are pretty expensive them alone since they require precision machinery and CNC metal working. The machines are also pretty expensive for normal business, so I don't think that any manufacturer is actually making them for hobby purposes.
However, if you are interested in little production and the only effect you wish to achieve is the bump on the paper, you should be able to look for some little, old pedal-powered letterpress machines from specialized antique trades; they should need metal types, however, which are harder to work with than one can immagine; another option is to look for a small letterpress machine which works with polymer plates. Polymer plates are cheap, affordable, they can emboss various shape and don't cost too much (there are even some on-demand services that make them with your designs) and are perfect for small editions. However, machinery that gives good results would make it a pretty expensive hobby.
Depending on where you live, you could look after a FabLab or Cooperative Workspace and see if they possess such a tool - this would be the best option since you could work at your project with an "hobbistic-friendly budget" without having to buy the whole thing.