Search/Filter Integration: Update Counter and Improve Visual Feedback

### Overview
We need to integrate the search bar with the filter system on the Projects page so that search terms update the filter counter, provide immediate visual feedback, and allow users to continuously add multiple search terms. Currently, users have no visual indication that their search was applied unless they manually open the filter bar, and they cannot add multiple search terms in a continuous workflow.

### Action Items
- [ ] Update filter counter to increment when search is performed (e.g., "Filters" → "Filters (1)")
- [ ] Make search terms immediately visible in the "Applied Filters" section without requiring the filter bar to be opened
- [ ] Enable continuous search functionality so users can add multiple search terms without reopening the filter bar
- [ ] Verify that the filter counter accurately reflects the total number of active filters AND search terms
- [ ] Test that all changes work correctly in both mobile and desktop views

### Resources/Instructions
- ER: Search Bar Not Persisting Results and Missing Filter Bar Integration #8420
* Technical consideration: The root problem is that the search bar and filter bar operate independently. The fix should make them work together as a single, unified system.
* Note: "Clear All" button currently works correctly and clears search results.