[api] Keep Home Assistant action fields in a flash table (#20237)

tests/integration/fixtures/api_string_lambda.yaml

@@ -37,7 +37,7 @@ api:
             args: [input_number]
 
         # Test that int lambdas still work correctly with to_string
-        # The TemplatableStringValue should automatically convert int to string
+        # Home Assistant action lambdas convert an int result to a string
         - homeassistant.event:
             event: esphome.test_int_lambda
             data:
@@ -54,7 +54,7 @@ api:
             args: [input_float]
 
         # Test that float lambdas still work correctly with to_string
-        # The TemplatableStringValue should automatically convert float to string
+        # Home Assistant action lambdas convert a float result to a string
         - homeassistant.event:
             event: esphome.test_float_lambda
             data:

tests/integration/test_api_string_lambda.py

@@ -1,4 +1,4 @@
-"""Integration test for TemplatableStringValue with string lambdas."""
+"""Integration test for Home Assistant action lambdas returning strings and numbers."""
 
 from __future__ import annotations
 
@@ -16,7 +16,7 @@ async def test_api_string_lambda(
     run_compiled: RunCompiledFunction,
     api_client_connected: APIClientConnectedFactory,
 ) -> None:
-    """Test TemplatableStringValue works with lambdas that return different types."""
+    """Test Home Assistant action lambdas that return different types are sent as strings."""
     loop = asyncio.get_running_loop()
 
     # Track log messages for all four service calls