docs: for FlowLiveUpdater to accomodate latest async/start change

docs/docs/core/flow_methods.mdx

@@ -1,5 +1,6 @@
 ---
 title: Flow Running
+toc_max_heading_level: 4
 description: Run a CocoIndex Flow, including build / update data in the target storage and evaluate the flow without changing the target storage.
 ---
 
@@ -70,8 +71,7 @@ This is to achieve best efficiency.
 
 ### One time update
 
-<Tabs queryString="code_lang">
-<TabItem value="shell" label="Shell" default>
+#### CLI
 
 The `cocoindex update` subcommand creates/updates data in the target storage.
 
@@ -81,7 +81,9 @@ Once it's done, the target data is fresh up to the moment when the function is c
 python main.py cocoindex update
 ```
 
-</TabItem>
+#### Library API
+
+<Tabs>
 <TabItem value="python" label="Python">
 
 The `update()` async method creates/updates data in the target storage.
@@ -108,8 +110,7 @@ A data source may enable one or multiple *change capture mechanisms*:
 
 Change capture mechanisms enable CocoIndex to continuously capture changes from the source data and update the target data accordingly, under live update mode.
 
-<Tabs queryString="code_lang">
-<TabItem value="shell" label="Shell" default>
+#### CLI
 
 To perform live update, run the `cocoindex update` subcommand with `-L` option:
 
@@ -120,7 +121,9 @@ python main.py cocoindex update -L
 If there's at least one data source with change capture mechanism enabled, it will keep running until the aborted (e.g. by `Ctrl-C`).
 Otherwise, it falls back to the same behavior as one time update, and will finish after a one-time update is done.
 
-</TabItem>
+#### Library API
+
+<Tabs>
 <TabItem value="python" label="Python">
 
 To perform live update, you need to create a `cocoindex.FlowLiveUpdater` object using the `cocoindex.Flow` object.
@@ -145,15 +148,19 @@ my_updater = cocoindex.FlowLiveUpdater(
 
 A `FlowLiveUpdater` object supports the following methods:
 
+*   `start()`: Start the updater.
+    CocoIndex will continuously capture changes from the source data and update the target data accordingly in background threads managed by the engine.
 *   `abort()`: Abort the updater.
-*   `wait()` (async): Wait for the updater to finish. It only unblocks in one of the following cases:
+*   `wait()`: Wait for the updater to finish. It only unblocks in one of the following cases:
     *   The updater was aborted.
     *   A one time update is done, and live update is not enabled:
         either `live_mode` is `False`, or all data sources have no change capture mechanisms enabled.
 *   `update_stats()`: It returns the stats of the updater.
 
 ```python
 my_updater = cocoindex.FlowLiveUpdater(demo_flow)
+# Start the updater.
+my_updater.start()
 
 # Perform your own logic (e.g. a query loop).
 ...
@@ -163,26 +170,47 @@ print(my_updater.update_stats())
 # Abort the updater.
 my_updater.abort()
 # Wait for the updater to finish.
-await my_updater.wait()
+my_updater.wait()
 ```
 
 Python SDK also allows you to use the updater as a context manager.
-It will abort and wait for the updater to finish automatically when the context is exited.
-The following code is equivalent to the code above:
+It will automatically start the updater during the context entry, and abort and wait for the updater to finish automatically when the context is exited.
+The following code is equivalent to the code above (if no early return happens):
 
 ```python
-async with cocoindex.FlowLiveUpdater(demo_flow) as my_updater:
+with cocoindex.FlowLiveUpdater(demo_flow) as my_updater:
     # Perform your own logic (e.g. a query loop).
     ...
     print(my_updater.update_stats())
 ```
 
-Within a synchronous function, remove `async` before `with`, like this:
+CocoIndex also provides asynchronous versions of APIs for blocking operations, including:
 
-```python
-with cocoindex.FlowLiveUpdater(demo_flow) as my_updater:
+*   `start_async()` and `wait_async()`, e.g.
+
+    ```python
+    my_updater = cocoindex.FlowLiveUpdater(demo_flow)
+    # Start the updater.
+    await my_updater.start_async()
+
+    # Perform your own logic (e.g. a query loop).
     ...
-```
+
+    # Print the update stats.
+    print(my_updater.update_stats())
+    # Abort the updater.
+    my_updater.abort()
+    # Wait for the updater to finish.
+    await my_updater.wait_async()
+    ```
+*   Async context manager, e.g.
+
+    ```python
+    async with cocoindex.FlowLiveUpdater(demo_flow) as my_updater:
+        # Perform your own logic (e.g. a query loop).
+        ...
+        print(my_updater.update_stats())
+    ```
 
 </TabItem>
 </Tabs>
@@ -191,8 +219,7 @@ with cocoindex.FlowLiveUpdater(demo_flow) as my_updater:
 
 CocoIndex allows you to run the transformations defined by the flow without updating the target storage.
 
-<Tabs queryString="code_lang">
-<TabItem value="shell" label="Shell" default>
+### CLI
 
 The `cocoindex evaluate` subcommand runs the transformation and dumps flow outputs.
 It takes the following options:
@@ -208,7 +235,9 @@ Example:
 python main.py cocoindex evaluate --output-dir ./eval_output
 ```
 
-</TabItem>
+### Library API
+
+<Tabs>
 <TabItem value="python" label="Python">
 
 The `evaluate_and_dump()` method runs the transformation and dumps flow outputs to files.