[codex] Add use_responses_lite 'override' logic (#26487)

## Summary

- add a defaulted `ModelInfo.use_responses_lite` catalog field
- support serializing `reasoning.context` while preserving the existing
effort and summary path
- has not been turned on for any models yet

I've added an override to parallel tools if responses_lite is on. I've
also forced persistent reasoning when using responses_lite. It would be
ideal if we could centralize all the responses_lite plumbing, but I
think this is best for now to keep the plumbing & diffs small.

## Testing

- `cargo test -p codex-protocol
model_info_defaults_availability_nux_to_none_when_omitted`
- `RUST_MIN_STACK=8388608 cargo test -p codex-core
responses_lite_sets_all_turns_context_and_disables_parallel_tool_calls`
- `RUST_MIN_STACK=8388608 cargo test -p codex-core
configured_reasoning_summary_is_sent`
- `cargo check -p codex-core --tests`
- `RUST_MIN_STACK=8388608 cargo clippy -p codex-core --tests` (passes
with pre-existing warnings in `codex-code-mode` and
`codex-core-plugins`)

codex-rs/core/src/client.rs

@@ -44,6 +44,7 @@ use codex_api::RawMemory as ApiRawMemory;
 use codex_api::RealtimeCallClient as ApiRealtimeCallClient;
 use codex_api::RealtimeSessionConfig as ApiRealtimeSessionConfig;
 use codex_api::Reasoning;
+use codex_api::ReasoningContext;
 use codex_api::RequestTelemetry;
 use codex_api::ReqwestTransport;
 use codex_api::ResponseCreateWsRequest;
@@ -609,6 +610,7 @@ impl ModelClient {
             reasoning: effort.map(|effort| Reasoning {
                 effort: Some(effort),
                 summary: None,
+                context: None,
             }),
         };
 
@@ -727,6 +729,11 @@ impl ModelClient {
                 } else {
                     Some(summary)
                 },
+                // When Responses Lite is disabled, omit context so Responses uses the default,
+                // which is currently `current_turn`.
+                context: model_info
+                    .use_responses_lite
+                    .then_some(ReasoningContext::AllTurns),
             })
         } else {
             None
@@ -775,7 +782,7 @@ impl ModelClient {
             input,
             tools,
             tool_choice: "auto".to_string(),
-            parallel_tool_calls: prompt.parallel_tool_calls,
+            parallel_tool_calls: prompt.parallel_tool_calls && !model_info.use_responses_lite,
             reasoning,
             store: provider.is_azure_responses_endpoint(),
             stream: true,