docs(postgres): configure db username/password separately

docs/docs/core/initialization.mdx

@@ -49,12 +49,29 @@ This takes care of the following effects:
 
 The following environment variables are supported:
 
-*   `COCOINDEX_DATABASE_URL`: The URL of the Postgres database to use as the internal storage, e.g. `postgres://cocoindex:cocoindex@localhost/cocoindex`
+*   `COCOINDEX_DATABASE_URL` (required): The URI of the Postgres database to use as the internal storage, e.g. `postgres://cocoindex:cocoindex@localhost/cocoindex`
+*   `COCOINDEX_DATABASE_USER` (optional): The username for the Postgres database. If not provided, username will come from `COCOINDEX_DATABASE_URL`.
+*   `COCOINDEX_DATABASE_PASSWORD` (optional): The password for the Postgres database. If not provided, password will come from `COCOINDEX_DATABASE_URL`.
 
 ## Explicit Initialization
 
 Alternatively, for flexibility, you can also explicitly initialize the library by the `init()` function:
 
+### Settings
+
+It takes a `Settings` object as argument, which is a dataclass that contains the following fields:
+
+*   `database` (type: `DatabaseConnectionSpec`, required): The connection to the Postgres database.
+
+#### DatabaseConnectionSpec
+
+`DatabaseConnectionSpec` has the following fields:
+*   `uri` (type: `str`, required): The URI of the Postgres database to use as the internal storage, e.g. `postgres://cocoindex:cocoindex@localhost/cocoindex`.
+*   `user` (type: `str`, optional): The username for the Postgres database. If not provided, username will come from `uri`.
+*   `password` (type: `str`, optional): The password for the Postgres database. If not provided, password will come from `uri`.
+
+### Example
+
 <Tabs>
 <TabItem value="python" label="Python" default>
 
@@ -63,7 +80,11 @@ import cocoindex
 
 def main():
   ...
-  cocoindex.init(cocoindex.Settings(database_url="postgres://cocoindex:cocoindex@localhost/cocoindex"))
+  cocoindex.init(
+      cocoindex.Settings(
+          database=cocoindex.DatabaseConnectionSpec(
+              uri="postgres://cocoindex:cocoindex@localhost/cocoindex"
+  )))
   ...
 
 ...

docs/docs/ops/storages.md

@@ -36,7 +36,9 @@ It should be a unique table, meaning that no other export target should export t
 
 The spec takes the following fields:
 
-*   `database_url` (type: `str`, optional): The URL of the Postgres database to use as the internal storage, e.g. `postgres://cocoindex:cocoindex@localhost/cocoindex`. If unspecified, will use the same database as the [internal storage](/docs/core/basics#internal-storage).
+*   `database` (type: [auth reference](../core/flow_def#auth-registry) to `DatabaseConnectionSpec`, optional): The connection to the Postgres database.
+    See [DatabaseConnectionSpec](../core/initialization#databaseconnectionspec) for its specific fields.
+    If not provided, will use the same database as the [internal storage](/docs/core/basics#internal-storage).
 
 *   `table_name` (type: `str`, optional): The name of the table to store to. If unspecified, will generate a new automatically. We recommend specifying a name explicitly if you want to directly query the table. It can be omitted if you want to use CocoIndex's query handlers to query the table.