But note also that an Emacs instance can open frames on multiple displays. This is not necessarily that hard to get if, for instance, you start with some X frames and then invoke emacsclient in a terminal in a way that causes it to open a character-cell frame there. window-system is documented as a terminal-local variable for this reason, and similarly the display-foo-p functions take an optional selector argument.
Which means that if you care about that, querying any of those once from your init file will not necessarily do what you want, and you should consider attaching to something like after-make-frame-functions. The global menu-bar-mode explicitly states that it applies to all current and future frames, too; toggle-menu-bar-mode-from-frame seems to be the per-frame version.
But if it's just cosmetic and you don't care about some variance in that case, then whatever. :-)