On my main machine Neovim is my go-to terminal editor. I find it much superior to Vim (I have also tweaked it a bit).
The fact is, I'm the only developer who will be connecting to the servers I talked about in the question, but there is a great chance that in 3 to 6 months I will be employing 2/3 developers that will work directly on the servers as part of their work.
If I knew for sure that I would be the only one using those servers, I would install Neovim and upload my configuration. But since I want to keep things simple and standard, I figured I would keep the text editor vanilla.
But the question still remains, should I use Vim or Neovim? I've been using standard Vim on the servers for now, as it's more widespread, but I'm pretty sure that Neovim would make Vim users feel right at home, while also providing a better experience.