There are a few more info that could be added nowadays like Display Protocol (Wayland/X11) and Display size. FastFetch does this but Neofetch is globally recognized.
Actually while neofetch detects pretty well I'm using alacritty:
Terminal: alacritty
Probably they learned $TERM is really meaningless if using screen or tmux, but fastfetch totally misses this and mistakenly shows screen as the terminal:
Terminal: screen
The only thing I like of fastfetch over neofetch is that it's faster, :) And yes the display missing, but I've never considered that something of much interest for such output... To me neofetch is just fine, and on terminal it gives you a more accurate answer... In the end is a matter of taste... But what it does is well done, :)
They don't run by themselves, they need a terminal emulator, or a console, underneath, so they can work. You can actually call screen on a console without graphical environment, and it'll provide the console all benefits of multiplexing. That doesn't make the multiplexer a terminal emulator by itself.
So, in my mind no, screen is not a terminal emulator, alacritty is, like xterm is, and so on. The multiplexor just adds extra capabilities to the terminal emulator.
At any rate, it's not worth going any further. What I meant is that neofetch was able to find out and show I'm using alacritty, whereas fastfetch doesn't show alacritty. And we can argue about the virtue of one or the other, but it'll boil down to taste. I prefer how neofetch shows alacritty, hehe. Some might prefer fastfetch showing screen. And most importantly, this is not critical at all.
There's an issue on fastfetch filed about it, and one of the devs indicated when using the screen multiplexer, they could find out the terminal emulator underneath, however they couldn't do the same with tmux. And to be consistent among multiplexers, they decided not to expose the terminal emulator underneath when using multiplexers, just show the multiplexer. I don't agree with that argument, but it's the dev right to choose to do that.