Why do you say the views are exclusive ? I repeat, you are toggling a search on your Inbox. Therefor, you are not modifying a property of the Welcome db, but of your Inbox.
What you call the "mini-mail view" is your Inbox, nothing else. It is appearing in the Welcome Page because the Welcome Page is a frameset with a frame dedicated to showing the Inbox of your mailbox. So, when you toggle search options in the Welcome Page, you are still changing properties of your Inbox.
Now, when you say that the change is random, that is interesting. Does that mean that sometimes you can deactivate the Search in the Inbox and it doesn't "show" in the Welcome Page (or vice-versa) ?
I think you will find that such a situation only happens when you have both the Welcome Page and the Inbox open. Since you always have the Welcome Page open (normally), that means more specifically that such "randomness" occurs only when your Inbox is open. That is because bth items are active, and not refreshing automatically.
I suggest that you make a test : start your Notes, watch the Welcome Page open and then open your Inbox. Note that the search bar option is identical in both views (it has to be). Toggle the search option in the Inbox, and close the view. You should note that the Welcome page has not changed. Close Notes. Restart, and note that the Welcome page search option is now again identical to the Inbox. The Welcome Page has been refreshed.
I think this is how it should happen. The changes only appear to be random. Actually it is a question of view refresh.
Merry XMas,
Pascal.