[Index Management] Use _count API instead of stats API for docs counts #239103
Description
The _stats API uses Lucene document counts, which includes nested documents. Vectors are stored as nested documents, which means that the docs count from that API is at least double what a vector search user would expect to see. The _count API only includes root documents, which is more natural for users to understand.
Currently, Index Management uses the _stats API to display docs counts. We should move that to the _count API so that docs counts shown on Index Management pages match what users expect to see.