Digital Ocean droplets can be created within your price range with a very vanilla version of Debian installed. My personal website runs off one. I've used AWS

Digital Ocean droplets can be created within your price range with a very vanilla version of Debian installed. My personal website runs off one. I've used AWS a lot for larger projects but they do a lot more to create vendor lock-in which breaks the idea of a "Linux server", i.e. it's easy to end up inadvertently depending on Amazon tools. Running your own physical server is also a valuable experience, but it's a significantly larger commitment which is less reward-per-effort in my opinion.