DOC: Clarify that file databases can also be written using to_file (#2821)

geopandas/geodataframe.py

@@ -1203,6 +1203,14 @@ individually so that features may have different properties
         With selected drivers you can also append to a file with `mode="a"`:
 
         >>> gdf.to_file('dataframe.shp', mode="a")  # doctest: +SKIP
+
+        Using the engine-specific keyword arguments it is possible to e.g. create a
+        spatialite file with a custom layer name:
+
+        >>> gdf.to_file(
+        ...     'dataframe.sqlite', driver='SQLite', spatialite=True, layer='test'
+        ... )  # doctest: +SKIP
+
         """
         from geopandas.io.file import _to_file
 
@@ -1902,6 +1910,10 @@ individually so that features may have different properties
         This method requires SQLAlchemy and GeoAlchemy2, and a PostgreSQL
         Python driver (e.g. psycopg2) to be installed.
 
+        It is also possible to use :meth:`~GeoDataFrame.to_file` to write to a database.
+        Especially for file geodatabases like GeoPackage or SpatiaLite this can be
+        easier.
+
         Parameters
         ----------
         name : str

geopandas/io/sql.py

@@ -112,6 +112,9 @@ def _read_postgis(
     Returns a GeoDataFrame corresponding to the result of the query
     string, which must contain a geometry column in WKB representation.
 
+    It is also possible to use :meth:`~GeoDataFrame.read_file` to read from a database.
+    Especially for file geodatabases like GeoPackage or SpatiaLite this can be easier.
+
     Parameters
     ----------
     sql : string
@@ -149,7 +152,7 @@ def _read_postgis(
 
     SpatiaLite
 
-    >>> sql = "SELECT ST_Binary(geom) AS geom, highway FROM roads"
+    >>> sql = "SELECT ST_AsBinary(geom) AS geom, highway FROM roads"
     >>> df = geopandas.read_postgis(sql, con)  # doctest: +SKIP
     """