Why does SSH require different paths for binaries (sudo without password)?

In this post: Btrfs backup tools in NixOS - #3 by symphorien I found the solution for running sudo commands without password.
Once @symphorien used { command = "${pkgs.btrfs-progs}/bin/btrfs"; options = [ "NOPASSWD" ]; }.
This is the version that I learned is the correct way to reference binaries but like he noted it doesn’t work over SSH.
For SSH we have to use this one here:
{ command = "/run/current-system/bin/btrfs"; options = [ "NOPASSWD" ]; }

He doesn’t mention why, however.

When you run a command without a full path, it will look into the PATH variable for the first match. Although binaries are stored in the nix store which is reflected by the value of the pkgs.btrfs-progs path, the actual path that your shell uses and the SUDO binary sees is a symlink in /run/current-system/sw/bin.

I think that btrbk uses the real physical paths when running locally but when running over SSH it has no way of knowing the real path so it just relies on the binaries being somewhere in the PATH environment variable directories.

Thank you sounds plausible.