In addition to threadi’s suggestion: that specific error is also a classic sign of a plugin registering a hook callback that no longer exists, so if the manual update doesn’t fix it, try deactivating all plugins via FTP (rename the wp-content/plugins folder) and see if the site comes back. If it does, rename the folder back and reactivate the plugins one by one to find the culprit.
You are right that the built-in search normally requires every search word to match (AND). When it behaves like an OR search, something is usually altering the query — often a search-related plugin or the theme’s own search handling. It is worth noting that Relevanssi itself defaults to OR logic, so what you have been seeing may have come from that rather than core search. A quick test is to temporarily switch to a default theme and turn off search-related plugins, then compare the results before re-enabling them one by one to find the culprit.
