> It is installed on virtually every UNIX-like system in the world
It's a vi command that is available "on virtually every UNIX-like system in the world", and that command may be provided by Vim (most Linux distributions and MacOS), nvi (BSDs), a limited reimplementation (BusyBox-based distributions), or the real vi itself (Arch). Even if you actually have Vim by default, the Vim you have is rarely suitable for extensive use so you end up installing a beefier build anyway.
The subset of vi that's common to all those implementations is what's on every UNIX-like system.
> Its modes are insert, normal, visual, search and command mode.
"Normal mode" and "command mode" are synonyms, you meant "command-line mode", you make that mistake again, later. There's no "search mode". Those are the main ones, there are a bunch of others.
A couple of observations, though…
> It is installed on virtually every UNIX-like system in the world
It's a vi command that is available "on virtually every UNIX-like system in the world", and that command may be provided by Vim (most Linux distributions and MacOS), nvi (BSDs), a limited reimplementation (BusyBox-based distributions), or the real vi itself (Arch). Even if you actually have Vim by default, the Vim you have is rarely suitable for extensive use so you end up installing a beefier build anyway.
The subset of vi that's common to all those implementations is what's on every UNIX-like system.
> Its modes are insert, normal, visual, search and command mode.
"Normal mode" and "command mode" are synonyms, you meant "command-line mode", you make that mistake again, later. There's no "search mode". Those are the main ones, there are a bunch of others.