Sorry not answering before.
Admin, you got the idea, this is the important bit. Also it seems you found an easy way to do it, and went further away seeing that this is useful in the statistics tab, well done. I really like the idea of filtering with more options.
Adding to this, how about allowing to save the filter parameters, so on the arrow on the right, one could select the filtering parameters saved before, and also jump to the advanced search?
However, to me it is still necessary to have the ability to search across multiple days, as the filter only works for the current day. In fact, the parameters for filtering could well be used in the search window as well.
The examples you provided are all clear to me. I would not change anything. In fact, I would like (and possibly this would be useful for other users as well), if there were more details about the filter parameters, such as all application reserved codes, how could they be used together (such as not, and, or, !, ~, comma, etc), the rules of use, and few examples, not that many. Not that they are not useful, they are, but for me the emphasis should be in defining the rules, not in guessing and discovering by ourselves. But if everything that can be done with the filter is already shown in the faq pages, then nothing else to add.