The terminal emulator is the window displaying the text. That used to be a physical screen called a terminal. The program that decides what text to show is the shell.

A terminal emulator (such as xterm, tmux, or emacs) is responsible for displaying a pseudoterminal in a different context (such as an x server, another terminal, or an emacs buffer respectively). Terminals are divided into three types: virtual terminals (on boot or as presented by [C-]A-F<N>), ‘true’ terminals (exposed on serial ports), and pseudoterminals (controlled by other programs). Essentially, you have a virtual terminal running X running Xterm which controls a pseudoterminal with tmux attached controlling another pseudoterminal.

Login shells are a special flag set when launching shells, sometimes. They typically read different settings (e.g. Bash reads ~/.bash_profile instead of ~/.bashrc). Most shells will launch as a login shell if their command name (argv[0]) begins with a hyphen; Bash also allows explicit bash -l to run a login shell.