Allow clearing filters and display entry filtered count

We already display the full entry count. We'd also like to show the filtered entry count.
This is potentially intertwined with a "clear filters" button, so both tasks are a single issue.

I think we want to differentiate between clearing filters and clearing the search text. But...I'm not sure.
Off the cuff, my favorite idea is:

• An X in the search box to clear the text
• The badge (see below) for seeing the filtered count and clearing it

I'm not convinced that's the ultimate solution though.

Some ideas for this were generated here:

I commonly see a count show on mobile apps when configuring filters. It's often combined with the close button e.g. "Show 434 matching results". The point is that the user can't configure filters and see the results loading in the background at the same time, so you give them an indication of what the current filters returned. I like that. On mobile we now use a drawer for the filters, which could align well with that pattern.

Start from the existing full entry count and the filter UI, then review the linked discussion for the proposed interaction patterns. Define how filtered-count display, filter clearing, and search-text clearing should differ, including the mobile filter drawer; done means the chosen behavior is implemented consistently.