Shell Terminology – What is an Interactive Shell?

I assume you're talking about Bash's concept of login vs. non-login and interactive vs. non-interactive Bash shells as described in the Invocation section of the manpage. (This is different from the interpretation in James Youngman's answer of any arbitrary command used as the "shell" (or user command interpreter) in the passwd(5) file and whether or not that program accepts user input; some, such as /usr/sbin/nologin, obviously don't.)

You are correct that /bin/bash --login some-script.sh will produce a non-interactive login Bash invocation, and this is perhaps a pathological example. There is one case, perhaps uncommon but not truly weird, that produces a non-interactive login shell: ssh somehost < some-file. Here sshd will start Bash with argv[0] set to -bash because it's not been given a command to run, causing Bash to consider itself a login shell, but, because stdin is not connected to a terminal, Bash will not set itself to interactive mode ($- will not contain i).

(Personally, that case seems far more reasonable than the converse, ssh somehost somecommand, which is not considered a "login shell" even though it's a fresh login to somehost just as the above is.)

I have recently done what I should have done long ago and put together a table of Bash's modes and what init files are run. If you're finding it confusing, take heart in that at least I do as well. It mystifies me what their original aim was with the rules about when .bashrc is executed.

When you log in to your x session via a display manager or in a tty, /etc/profile is (usually - apparently it is being in your case, though some graphical shells do not read it) sourced by your shell program. After that, a local file (I'm assuming you're using bash here) ~/.bash_profile, ~/.bash_login or ~/.profile will be sourced, and any environment variables defined here will override /etc/profile for the current user. This environment is inherited by any shell you open within the session. This is why we can define environment variables, such as your PATH, in these files.

When you open gnome-terminal, yes by default that starts a non-login shell, but it inherits your user environment already loaded from the login shell or graphical shell. Since ~/.bashrc is sourced when starting an interactive shell (eg opening gnome-terminal), it may be used to override some elements of the environment (such as PS1).

(gnome-terminal itself is an application, not a shell)