Set verbosity to low for 5.1 (#6568)

And improve test coverage

codex-rs/core/src/client.rs

@@ -215,7 +215,9 @@ impl ModelClient {
         let input_with_instructions = prompt.get_formatted_input();
 
         let verbosity = if self.config.model_family.support_verbosity {
-            self.config.model_verbosity
+            self.config
+                .model_verbosity
+                .or(self.config.model_family.default_verbosity)
         } else {
             if self.config.model_verbosity.is_some() {
                 warn!(

codex-rs/core/src/model_family.rs

@@ -1,3 +1,5 @@
+use codex_protocol::config_types::Verbosity;
+
 use crate::config::types::ReasoningSummaryFormat;
 use crate::tools::handlers::apply_patch::ApplyPatchToolType;
 use crate::tools::spec::ConfigShellToolType;
@@ -55,6 +57,9 @@ pub struct ModelFamily {
     /// If the model family supports setting the verbosity level when using Responses API.
     pub support_verbosity: bool,
 
+    // The default verbosity level for this model family when using Responses API.
+    pub default_verbosity: Option<Verbosity>,
+
     /// Preferred shell tool type for this model family when features do not override it.
     pub shell_type: ConfigShellToolType,
 }
@@ -78,7 +83,9 @@ macro_rules! model_family {
             effective_context_window_percent: 95,
             support_verbosity: false,
             shell_type: ConfigShellToolType::Default,
+            default_verbosity: None,
         };
+
         // apply overrides
         $(
             mf.$key = $value;
@@ -174,6 +181,7 @@ pub fn find_family_for_model(slug: &str) -> Option<ModelFamily> {
             supports_reasoning_summaries: true,
             apply_patch_tool_type: Some(ApplyPatchToolType::Freeform),
             support_verbosity: true,
+            default_verbosity: Some(Verbosity::Low),
             base_instructions: GPT_5_1_INSTRUCTIONS.to_string(),
         )
     } else if slug.starts_with("gpt-5") {
@@ -202,5 +210,6 @@ pub fn derive_default_model_family(model: &str) -> ModelFamily {
         effective_context_window_percent: 95,
         support_verbosity: false,
         shell_type: ConfigShellToolType::Default,
+        default_verbosity: None,
     }
 }