docs: fix async pc.index() usage — it's a coroutine, must be awaited

AsyncPinecone.index() is a coroutine that resolves to AsyncIndex, but
several examples treated it as a synchronous factory. Every existing
async sample either assigned the coroutine to a local without awaiting
it (then crashed on the first method call with "'coroutine' object
has no attribute upsert/query"), or used `async with pc.index(...) as
index:` directly on the coroutine (which is also wrong — async with
needs the resolved AsyncIndex, not the coroutine).

Same bug we hit in the harness's _build_client_async_sdk2 earlier
today.

Fixes:
- README.md async quickstart: add `await` to `pc.index(host=...)`
- sync-vs-async.md: rewrite the prose ("pc.index() is a synchronous
  factory method — it does not need to be awaited") which had it
  exactly backwards on the async client; switch every async example
  to the two-step pattern `index = await pc.index(...); async with
  index: ...` for readability

Sphinx build verified clean after the change.

Co-Authored-By: Claude Opus 4.7 (1M context) <noreply@anthropic.com>

Author: jhamon

README.md

@@ -67,7 +67,7 @@ from pinecone import AsyncPinecone
 async def main():
     async with AsyncPinecone(api_key="your-api-key") as pc:
         desc = await pc.indexes.describe("movie-recommendations")
-        index = pc.index(host=desc.host)
+        index = await pc.index(host=desc.host)
         async with index:
             results = await index.query(
                 vector=[0.012, -0.087, 0.153],  # 1536-dim vector

docs/guides/sync-vs-async.md

@@ -53,22 +53,23 @@ from pinecone import AsyncPinecone
 async def main() -> None:
     async with AsyncPinecone() as pc:
         desc = await pc.indexes.describe("product-search")
-        index = pc.index(host=desc.host)
+        index = await pc.index(host=desc.host)
 
         await index.upsert(vectors=[("product-42", [0.1, 0.2, ...])])
         results = await index.query(vector=[0.1, 0.2, ...], top_k=5)
 
 asyncio.run(main())
 ```
 
-`pc.index()` is a synchronous factory method — it does not need to be awaited. It
-returns an `AsyncIndex` that you use with `await`. The `AsyncIndex` itself manages its
-own HTTP session; close it with `async with index:` or `await index.close()`.
+`pc.index()` on `AsyncPinecone` is itself a coroutine — `await` it to get back an
+`AsyncIndex`. The returned `AsyncIndex` manages its own HTTP session; close it
+with `async with index:` or `await index.close()`.
 
 ```python
 async with AsyncPinecone() as pc:
     # Preferred: pass the host directly to skip a describe call
-    async with pc.index(host="product-search-abc123.svc.pinecone.io") as index:
+    index = await pc.index(host="product-search-abc123.svc.pinecone.io")
+    async with index:
         results = await index.query(vector=[0.1, 0.2, ...], top_k=5)
 ```
 
@@ -108,7 +109,8 @@ async def main() -> None:
     async with AsyncPinecone() as pc:
         # Resolve the host first, then create the index client
         desc = await pc.indexes.describe("movie-recommendations")
-        async with pc.index(host=desc.host) as index:
+        index = await pc.index(host=desc.host)
+        async with index:
             await index.upsert(vectors=[
                 ("movie-42", [0.012, -0.087, 0.153, ...]),
                 ("movie-99", [0.045,  0.021, -0.064, ...]),
@@ -151,7 +153,8 @@ import asyncio
 from pinecone import AsyncPinecone
 
 async def upsert_batch(pc: AsyncPinecone, batch: list[tuple[str, list[float]]]) -> None:
-    async with pc.index(host="product-search-abc123.svc.pinecone.io") as index:
+    index = await pc.index(host="product-search-abc123.svc.pinecone.io")
+    async with index:
         await index.upsert(vectors=batch)
 
 async def main() -> None:
@@ -191,7 +194,8 @@ each `AsyncIndex` separately:
 
 ```python
 async with AsyncPinecone() as pc:
-    async with pc.index(host="...") as index:
+    index = await pc.index(host="...")
+    async with index:
         await index.query(...)
     # index is closed here; pc is closed at the outer block exit
 ```