Search Box in the chrome works differently then any other place in the product. If you look at default.master(%Program Files%\Common Files\Microsoft Shared\Web Server Extensions\12\TEMPLATE\GLOBAL) file you can find the following
So basically the search box is hosted by delegate control. Essentially the idea here is that WSS comes with a search box which supports certain features. When you install MSS or MOSS a newer Search box feature is installed with higher priority) and the delegate control know to pick this up. You can read more about the delegate controls here:
Create a new feature that will put the search box with correct properties that you want: Follow the steps in
In step 3 in the linked article instead of using what is specified in the article use this
<?xml version="1.0" encoding="utf-8" ?>
ControlClass="Microsoft.SharePoint.Portal.WebControls.SearchBoxEx" ControlAssembly="Microsoft.SharePoint.Portal, Version=18.104.22.168, Culture=neutral, PublicKeyToken=71e9bce111e9429c">
You do not need to do step 4 and 5 in the linked article. Note that I have changed the sequence number to 24 which will ensure that your control is what the delegate control render. It should be between 0-25 as the OOB search box has a sequence 25.
The property that I added is highlighted in green.
The value is ShowDD by default if nothing is set.