Closes #43408

Previously, we checked the setting inside `is_completion_trigger()`,
which only affects LSP completions. This was ok because user-defined
snippets were tacked onto LSP completions. Then #42122 and #42398 made
snippet completions their own thing, similar to word completions,
surfacing #43408. This PR moves the settings check into
`open_or_update_completions_menu()` so it applies to all completions.

Release Notes:

- Fixed setting `show_completions_on_input: false` so that it affects
word and user-defined snippet completions as well as LSP completions